{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0001","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3b688840eb5a6621ae3dee0bd6af91adb4f35350605d9139efa54142142db48","family_id":"qf-mod34-rank-exact-rref","seed":202611031001,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,0],[0,1]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0002","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"113ea161e51f3e1bf1faa791cbaeda78a251c65812ca770ab82513db7ba45426","family_id":"qf-mod34-rank-exact-rref","seed":202611031002,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,1],[0,2]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0003","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b20c92ef86a0ccc747915a2c423e932caeabfad6af9db366eef660333f21ca1","family_id":"qf-mod34-rank-exact-rref","seed":202611031003,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,2],[0,3]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0004","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eedafa3241332e2c10712ace157789069d58950f2ebdbb72c7aef48bdf6526b","family_id":"qf-mod34-rank-exact-rref","seed":202611031004,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,3],[0,4]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0005","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39e4f2d3273fd71b638dbb2b592a26659a62c28a27e0484c1c5fd3570162c946","family_id":"qf-mod34-rank-exact-rref","seed":202611031005,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,4],[0,5]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0006","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94a461f853b9141620a37e6e6bb175fb95176873d9860dfbcfad525f4c5ebaf7","family_id":"qf-mod34-rank-exact-rref","seed":202611031006,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,5],[0,6]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0007","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d94ce833cc9e15df7d3bf0596d53db84bbffe42d146b8a279fb01708e2a66c","family_id":"qf-mod34-rank-exact-rref","seed":202611031007,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,6],[0,7]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0008","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11f4c8bc72db83c2063f2950b99555c6dd613931ed813755631c05dd5f3922ec","family_id":"qf-mod34-rank-exact-rref","seed":202611031008,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,7],[0,8]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0009","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c1ed33f2e2a8fb77774ea02d5b06dcb0b60741715a8e65d712a5d6e54935a23","family_id":"qf-mod34-rank-exact-rref","seed":202611031009,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,8],[0,9]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0010","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12c4847b182732ca4850203b43102cba631d277994a39d21f05a42d8bbc46e5b","family_id":"qf-mod34-rank-exact-rref","seed":202611031010,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,9],[0,10]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0011","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f4ca868848c8c65cdb3bab4d150b58fd011e9999bfb62649d8a5e5c5371414","family_id":"qf-mod34-rank-exact-rref","seed":202611031011,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,10],[0,11]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0012","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20a9ba4cb649920b6893abd1798cbd9316c7e80582a1f2779969cf385d899967","family_id":"qf-mod34-rank-exact-rref","seed":202611031012,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,11],[0,12]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0013","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5a8b157d1f06c17641d9108f808e58a710ce086ddf937983e88bfca343142e","family_id":"qf-mod34-rank-exact-rref","seed":202611031013,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,12],[0,13]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0014","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"203bc2b1d9ab468de370f37be7b6e0a750de8e0f33c76ff15fdb45cda9576f20","family_id":"qf-mod34-rank-exact-rref","seed":202611031014,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,13],[0,14]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0015","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb653d776a251df03fe43fe2ebf09daf4b9f05b936c60e0317a7124f026d38b","family_id":"qf-mod34-rank-exact-rref","seed":202611031015,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,14],[0,15]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0016","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41638eb8de98dd98db6e0186b1041f98751e9946b147eda7307bb18163cd096f","family_id":"qf-mod34-rank-exact-rref","seed":202611031016,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,15],[0,16]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0017","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ade343121687225fb6dccf975f0907c5f6ca7869ac5a67e6a8022050a2413b1b","family_id":"qf-mod34-rank-exact-rref","seed":202611031017,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,16],[0,17]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0018","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67d7de2f16f2260bc4b7f6ff633515487177a54e600ca96ab29857c7655680b","family_id":"qf-mod34-rank-exact-rref","seed":202611031018,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,17],[0,18]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0019","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab9118be7ce4f23d1c9a4e7a04a5518512772bee9908eaa36d8cbda841103b79","family_id":"qf-mod34-rank-exact-rref","seed":202611031019,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,18],[0,19]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0020","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"861d2916b69ab4ec4e7bbe9b9408d3c2bfc9dd6c8f26a3b9bb5bf00e067a4260","family_id":"qf-mod34-rank-exact-rref","seed":202611031020,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,19],[0,20]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0021","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19884383f71c08db0b8e7e9944c1547cd9e3ca6ff9de2574cc95ef5e7a3c7f70","family_id":"qf-mod34-rank-exact-rref","seed":202611031021,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,20],[0,21]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0022","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30d2684a384bcce7431d49557724ce21e126163b6fb8fc89742dfd0ed4920d59","family_id":"qf-mod34-rank-exact-rref","seed":202611031022,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,21],[0,22]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0023","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceaed207567bd864435a195f954de6042ab859952d2b9749f190229278a00683","family_id":"qf-mod34-rank-exact-rref","seed":202611031023,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,22],[0,23]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0024","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9736e72b0ebfb15a346fb20ad302d57a5546fd603afae203d3470756a2e4c6e0","family_id":"qf-mod34-rank-exact-rref","seed":202611031024,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,23],[0,24]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0025","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a6113d9808ef1d44579faa320c98d78eb7b117bb0ad935b724c64d5cfcd3191","family_id":"qf-mod34-rank-exact-rref","seed":202611031025,"semantic_template":"qf-mod34-rank-exact-rref/{\"matrix\":[[1,24],[0,25]]}"},"fields":{"title":"The Complete Exact RREF Supplies Matrix Rank","stem":"Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.","answer":{"kind":"expression","value":"([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)","explanation":"Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)."},"skills":["The Complete Exact RREF Supplies Matrix Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0026","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe11ca4e9935b683273aa4e1fdc47471e967404992ee6c1c749d34403d3c7bd3","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031026,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,0],[0,1]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0027","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7bbfc07463963f1fc697461da2e2ed44289706fd1945a8e021e6fc4032acf4d","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031027,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,1],[0,2]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0028","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2d6076a97d05dc307c57ccd9fbcc963f327a63148f2991c514e5f65f2da4bb","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031028,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,2],[0,3]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0029","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8abb94e1ddb5d592b87db8bf9bf8880ec2f7886cc5af6d9f8e00dd553131fce","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031029,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,3],[0,4]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0030","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"852856264bb45b932793edd35296d3cf0ba5b31018fa1d311b4b9373c27b6ec1","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031030,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,4],[0,5]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0031","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee61582e8a481aee52e81a0ecdc819fbcf8303797c8535b2e12893b6f45062d","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031031,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,5],[0,6]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0032","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d889a207d04fb479d9c4d8046eb2a9732685a9d8c2d412a8c03c80df0d90d5fd","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031032,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,6],[0,7]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0033","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5cb2e197ceef3a06b0fbb69d29a52e11dee213ca1f63f16c7f9444d21fd6ce2","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031033,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,7],[0,8]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0034","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b040d00efdad05887d2ed259701a83055786cd085c9783d3d83527acc25e130","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031034,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,8],[0,9]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0035","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ec1f8b332486cfad153ae2cb5c706b7da84e8950ac68a96d77b6438d58046d","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031035,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,9],[0,10]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0036","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fa306fa7476c4c09e56e84ae8d68c23e32a6c87ed41e663bb022f7a18310182","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031036,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,10],[0,11]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0037","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a8e59ba60602177bdf3122af895edae6dcabfead2140990d2c077c63fdff173","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031037,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,11],[0,12]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0038","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9412f62ac89658bbc1abe7101deba69629b05f96eac67c27771fcbc296b3514","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031038,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,12],[0,13]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0039","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"088bd3739510caf60cdb335ac6a770fa15a3a2d110bc0f7f17acd955674c0006","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031039,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,13],[0,14]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0040","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6acdccdae4399c150164f6a3991b91e4c1787110557cde9cbabd86845f21ba88","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031040,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,14],[0,15]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0041","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"419d5acd4560fd39f00d34f92f360b9a29682754d24a09f4bd06063f4f156367","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031041,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,15],[0,16]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0042","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0aa57db161d630ab78ee4a93231a7d05ce20ca54143926d4bb58b1369b8afcb","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031042,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,16],[0,17]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0043","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353fac346ca753e3c0bda82d353fafcf272001db62ce2c20d5c627f6c14f8a5a","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031043,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,17],[0,18]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0044","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3c059552755e329b6948e6902f511a5fcdef8026ce6147bd0dbbe3c0ee7282","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031044,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,18],[0,19]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0045","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3921c239915e83ff14df06a6b413c4dce28b36bd8d512445148e8477ccd987dd","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031045,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,19],[0,20]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0046","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54dba663e31066107ceb0d023ae41f7ab03f11546c4e6ec2f0dc0ef0de483318","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031046,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,20],[0,21]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0047","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5268414ebe37db3ea4017bff9027946317cec7209201230d2f99d0066664a6a5","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031047,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,21],[0,22]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0048","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3e23b06e0c14e0d35ca78b681c44fcf5e0f586d69f885120c09bde27b0d7a4b","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031048,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,22],[0,23]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0049","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3bb857e125a506a806f794b44771e6cf4a5d309f47cf8cbfc710bbeace216d9","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031049,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,23],[0,24]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0050","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b03e33b4576185631a0b675c1d7aef6f75bd7f619dc0b5854e14362892891185","family_id":"qf-mod34-rank-minor-lower-bound","seed":202611031050,"semantic_template":"qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,24],[0,25]],\"rows\":[0]}"},"fields":{"title":"A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank","stem":"Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).","answer":{"kind":"expression","value":"([0], [0], [[1]], 1, Yes, rank_at_least_1)","explanation":"Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."},"skills":["A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0051","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b534dd383cf728cfedf632fb16c70a99592ab00cc2c879575ef2364e41dbcbd","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031051,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,0],[0,1]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 1, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 1, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0052","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de4a2a26d0814046d9d7f5b2a81b69273e09407c93616901c2263aaf8313ea41","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031052,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,1],[0,2]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 2, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 2, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0053","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7cf86ad84016db4ca1b4a9af665d9509764acac8e08966eabd2b6fa3b43e015","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031053,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,2],[0,3]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 3, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 3, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0054","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44bf14451a19ef981342ea47ab31135b1d38be2517c17ffe5111f55a1191ca56","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031054,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,3],[0,4]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 4, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 4, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0055","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"666b89be81aa972878f6d979dfbe5dc96959045572d096c5befcbc85fbb2ce21","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031055,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,4],[0,5]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 5, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 5, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0056","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c121941aa8f4774f081590f6fca453edf20330f98a41a244d1679804f075a962","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031056,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,5],[0,6]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 6, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 6, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0057","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c49d00674978e80a9ec557702515a7fa16ca0dc35f93c81bf3039608cf1d41f9","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031057,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,6],[0,7]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 7, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 7, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0058","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0cd2e09405e501dc794d0f0d6e4138116e99a8b7d025a698450d7782150942d","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031058,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,7],[0,8]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 8, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 8, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0059","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb95b86e1af298544b7452e6ed011ec8c6cb8538409307fab531e9bf78ac8540","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031059,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,8],[0,9]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 9, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 9, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0060","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a92e37d8be1fe2b704484b2d7da8f5fd42610faf07af56f3c819a4efdf8eb8eb","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031060,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,9],[0,10]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 10, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 10, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0061","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d006793c81c930a4cec35f0b6ea7a4ed5609d5dbd32165c0526ddb83a24b03ef","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031061,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,10],[0,11]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 11, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 11, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0062","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca16331011b7c13a1da46b9066595750b70d8a9e1b652c1573c9d4f1c837481d","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031062,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,11],[0,12]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 12, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 12, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0063","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2541595851a451db8ac9f96f27fedca11b528f534bb53c1ac03d9d73eb2d3480","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031063,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,12],[0,13]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 13, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 13, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0064","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4eca65deef6720d01fda6e75416d52ec6e59b8e85b6315b920e33ab6e9da381","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031064,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,13],[0,14]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 14, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 14, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0065","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83eeb0f2f29c4e06f5ab99454d0f2a4ffe49a81c3bb6bd233c1d3dbdd3bdd407","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031065,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,14],[0,15]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 15, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 15, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0066","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da4eb68bef0b543f60548845f05c1a2bd89c085df350bdb4d475323fc8653ee","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031066,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,15],[0,16]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 16, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 16, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0067","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ca81dc9f1f298c3acf30982feafea22ff50567102bc202b70271bbbc7aeff8d","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031067,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,16],[0,17]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 17, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 17, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0068","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dce422bc12a471d547bfa0c6273af299e02c5545bec83cd7e106859555a37cbd","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031068,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,17],[0,18]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 18, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 18, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0069","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81072830162d204ac433b235884c178c12525c2d1c05438ed0241a247c5ebff8","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031069,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,18],[0,19]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 19, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 19, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0070","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2db9fa4577c9838575e391f43180ad4994be9155c30cc3e144046e0b920194e","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031070,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,19],[0,20]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 20, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 20, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0071","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"329dab7013b03da4e6ae46177775c1d91d95d5a89438474d377d0ace64e6a10f","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031071,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,20],[0,21]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 21, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 21, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0072","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739580aa3daf7ede9f51bba22a8df8cc8e14c93de4abc43ece0b60b7a47fd92c","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031072,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,21],[0,22]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 22, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 22, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0073","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5384ff9a14a84ed1a7a4c6005f587987a5fcb5fc01c678c695936d75148db4f","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031073,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,22],[0,23]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 23, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 23, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0074","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"712f279ad882e17384f545cd5f517b0c41c5f2ac47f8e7b1e94f34af16847dd1","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031074,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,23],[0,24]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 24, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 24, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0075","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"508113cdb41062eb39ec9a2b905752ba8a62a443950b75976ef06a1c7219226e","family_id":"qf-mod34-rank-full-rank-audit","seed":202611031075,"semantic_template":"qf-mod34-rank-full-rank-audit/{\"matrix\":[[1,24],[0,25]]}"},"fields":{"title":"Full Rank Needs the Correct Square-Matrix Certificate","stem":"Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).","answer":{"kind":"expression","value":"(2, 25, Yes, 2, Yes)","explanation":"Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 25, Yes, 2, Yes)."},"skills":["Full Rank Needs the Correct Square-Matrix Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0076","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"072a204c83694c72d914128127093f4644df928edff5ae0670c50be22356f3e9","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031076,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,1],[2,2]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 1], [2, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0077","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b32df5c6206a0da2592b44b202d8e79c5da9ca6fb1f021446eea979f689d714","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031077,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,2],[2,4]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 2], [2, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0078","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3634edbb5da1b6354f306f24c4498fa27ec825ba5a69143ffc0a88459496c824","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031078,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,3],[2,6]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 3], [2, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0079","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"943c0b2f0424cd202558c3cb9e6f7312b67441bdfa61a1f23657f254faee56cc","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031079,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,4],[2,8]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 4], [2, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0080","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52a48b1f177c0d614abf9ab8a56f2c749b42c5e9ff54e34ada9529a522cd7a30","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031080,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,5],[2,10]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 5], [2, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0081","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5ead9a78b5d18d45333e3ea3f88061e3894d680f1d769cf3782a92a15485c32","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031081,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,6],[2,12]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 6], [2, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0082","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b563c4432d1c57979db92783a2b463bbeb88d4a03b2a87ceca727382a6da9d1","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031082,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,7],[2,14]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 7], [2, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0083","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aca754efd918cc91bdb61fb12da6774bdcd0905002185244709127ea10f4aa3","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031083,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,8],[2,16]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 8], [2, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0084","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757511d38c795199e6c93489b243f547feb1f911e81a640574d4d194f3e5fe63","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031084,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,9],[2,18]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 9], [2, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0085","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a27c9341c9030122abf6be5037e95165efb7e7628b5c0d32d0b0d5b4fd5a1a81","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031085,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,10],[2,20]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 10], [2, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0086","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8fef617f4e228e87f69377dc2db1d30d0a37fc4885ded9c7b412da889961711","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031086,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,11],[2,22]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 11], [2, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0087","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30b99aaa50e8464415eb0adbda503a6f258e33362054dc2dc2b2f4979e19739f","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031087,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,12],[2,24]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 12], [2, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0088","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4ca8616408146350e319f95b23883ec368fac5850d6b96b506267868b04b9d","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031088,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,13],[2,26]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 13], [2, 26]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0089","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4e33a0e278cef246fd8f02b9114ec1580551f432cde199e5b181be75e11024","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031089,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,14],[2,28]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 14], [2, 28]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0090","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f6d4904509fc5186a39040cd9ed2ce2e3cdce3ebcdffbfdaccbbb252f6cd8fe","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031090,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,15],[2,30]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 15], [2, 30]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0091","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c90901f217ff1c14775d694dd7c3a62aeb6617a5fad51d1417ab342c723e4c8","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031091,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,16],[2,32]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 16], [2, 32]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0092","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc3e4e90bcf14f3c5fb67d301b87e9354da69f59300e0ecb62b97f2b0424580f","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031092,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,17],[2,34]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 17], [2, 34]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0093","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb9da64f5061499d12030a5ff00911cd48673577bd3476d5eb707aa979aa01dc","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031093,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,18],[2,36]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 18], [2, 36]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0094","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a8c8f796e009e41948d575697cab1ef3003905addcedaca03305b18dc1be724","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031094,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,19],[2,38]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 19], [2, 38]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0095","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e03983898ef0f2eca704e1c07540974837dd787daea3d176d059865b58be17f","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031095,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,20],[2,40]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 20], [2, 40]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0096","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a392bbbfb4d8869657995804e9e1e5f542a345ea078b3b8e6a71fefe90e61107","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031096,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,21],[2,42]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 21], [2, 42]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0097","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61c1556a831cbda48808150789259b613d4f4b30e28be0a1da5b8336d48ccd98","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031097,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,22],[2,44]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 22], [2, 44]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0098","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45af7cc14fa537debc5322096b8a82cb78ea9811581b00b65b7caedcab1bfdbd","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031098,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,23],[2,46]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 23], [2, 46]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0099","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6a24f0525917d411a6c0265407bb7f358a23a8cac79fcf5cf50366bcda402fb","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031099,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,24],[2,48]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 24], [2, 48]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r01-matrix-rank-0100","batch_id":"batch-20261014-advance-20261103-r01-matrix-rank-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41071f95ccc32a091e7a3f983d452ea3094e8e8445a5eddad099e179f45ebbab","family_id":"qf-mod34-rank-determinant-zero-boundary","seed":202611031100,"semantic_template":"qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,25],[2,50]]}"},"fields":{"title":"Zero Determinant Does Not Determine Which Lower Rank Occurs","stem":"Use the complete ordered integer matrix A=[[1, 25], [2, 50]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.","answer":{"kind":"expression","value":"(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)","explanation":"Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."},"skills":["Zero Determinant Does Not Determine Which Lower Rank Occurs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}