{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r04-determinant-0001","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c4e1ecfe0ebfe3d36a691050e0e5bb4090e6e099efe95c2c69767a4fbf44f26","family_id":"qf-density16-det-row-swap-signed","seed":202610164001,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[0,0],[0,0]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0002","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3e802ae5479641475b2ade015f9766f0c2411c070b4edadf80e36c67514e9a7","family_id":"qf-density16-det-row-swap-signed","seed":202610164002,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[1,2],[3,4]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0003","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62387838f9f9f3fb1601bc9ae20dee4548a915df791cdefcacd21f073b652a81","family_id":"qf-density16-det-row-swap-signed","seed":202610164003,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-10,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-10, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0004","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"174167591de709425edb091cbdc1facc3a2db4a1aed4703b4f035f669ddb1597","family_id":"qf-density16-det-row-swap-signed","seed":202610164004,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-9,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0005","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8869e9223c95144e11e5ba86b35407b7365394ad83c2a727a54bece8bcd11c49","family_id":"qf-density16-det-row-swap-signed","seed":202610164005,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-8,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0006","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46c6c7750f04e8d78aacddadb52e6fbfba6a1984aba963f4f493c47ec0174432","family_id":"qf-density16-det-row-swap-signed","seed":202610164006,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-7,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0007","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b21d3a7b62395c0b32246a616324f7e45e82a5c25b23267277fe004b223c90","family_id":"qf-density16-det-row-swap-signed","seed":202610164007,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-6,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0008","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e329a531bba6394155da8c3159c26823b02c9d49ab67abdd23dee8fce20ef3a","family_id":"qf-density16-det-row-swap-signed","seed":202610164008,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-5,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0009","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1081e97dded436aec0795a289256b3b5e9427ba78e98cae59d63832a2f48386","family_id":"qf-density16-det-row-swap-signed","seed":202610164009,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-4,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0010","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6a238f751864aaf63b5a325781c2c7d6ceacb19f3a25b9726961c9edbc843d","family_id":"qf-density16-det-row-swap-signed","seed":202610164010,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-3,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0011","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a3af2e316c2063e6866c81b41790ecebdcf23fbea1743314e4eaa304f4d8e2d","family_id":"qf-density16-det-row-swap-signed","seed":202610164011,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-2,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0012","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7757f6837d8d7f01ffbb43877fb4f7ff148cfaee1e87e3a6fd32df55f1b5c970","family_id":"qf-density16-det-row-swap-signed","seed":202610164012,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-1,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0013","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a10ac3a7d7d743622845865722c1890075d716225f8065d4e7cc2568cd3c153","family_id":"qf-density16-det-row-swap-signed","seed":202610164013,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[0,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0014","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac74afb8e8d99d925455fed8bc5250dcb6aea0a67b2cf76b705795a33f8fb4e6","family_id":"qf-density16-det-row-swap-signed","seed":202610164014,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[1,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0015","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f7eec22093cef53f0c1970cb3f17c255f34a2b98f2f31fa37d7b4cf18dcf4f1","family_id":"qf-density16-det-row-swap-signed","seed":202610164015,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[2,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0016","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cbc7ec721ffecc6a263be94ea6ad979fa04ecfb677bf6d6ae100c7b91b7e40a","family_id":"qf-density16-det-row-swap-signed","seed":202610164016,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[3,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0017","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5f01ce6f43cb84e0927633c74e4a3fc6d8c3ab72c6c8183e6837fef4af9bb7","family_id":"qf-density16-det-row-swap-signed","seed":202610164017,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[4,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0018","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590df254c900d90ebcf33b6475d899da17b5c3dab6423445c5d6da5110c8d072","family_id":"qf-density16-det-row-swap-signed","seed":202610164018,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[5,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0019","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48416d897a717474b42d7027c59139c89bd706dbc669b1728bc8fbb088e47540","family_id":"qf-density16-det-row-swap-signed","seed":202610164019,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[6,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0020","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9794883297c2d89064a7c9a2f23ed7cf5a4a4bcb5e3c7ed94031d4df1ef2b556","family_id":"qf-density16-det-row-swap-signed","seed":202610164020,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[7,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0021","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90c842bc53289c5d02b7c1b52a981f819c7d0ee55f517a0361862652bafeb67a","family_id":"qf-density16-det-row-swap-signed","seed":202610164021,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[8,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0022","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d6fef70682c7f6dc20534cbfd6aeb5d0935c9abb046b256f17e4afabb30caf0","family_id":"qf-density16-det-row-swap-signed","seed":202610164022,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[9,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0023","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b50f0077db35fe70f1e47183dda9de64d1d5a8861b31e881d624d92340a2e2f9","family_id":"qf-density16-det-row-swap-signed","seed":202610164023,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[10,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0024","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"386a0aeb2f0101066aaeb7f85bb9f06a4c83b56e41bb77b891e73dacb89362c8","family_id":"qf-density16-det-row-swap-signed","seed":202610164024,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[11,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0025","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c32d834498abed343c5c8aa7b716a62182dbac0ed3e3b05d1b18016a7d0578","family_id":"qf-density16-det-row-swap-signed","seed":202610164025,"semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[12,1],[2,1]]}"},"fields":{"title":"Signed Determinants Before and After a Row Swap","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)","explanation":"Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)."},"skills":["Signed Determinants Before and After a Row Swap"],"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-20261012-advance-20261016-r04-determinant-0026","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd982f8bc588b12f971eece8ceef2ba77d98f2371639bb526daf4d2eb5fe6ecf","family_id":"qf-density16-det-scaled-matrix","seed":202610164026,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[0,0],[0,0]],\"t\":0}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], t=0. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(0, [[0, 0], [0, 0]], 0, 0)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [[0, 0], [0, 0]], 0, 0)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0027","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afc3e096b01fae6a82f50f54678bbfbdf0ea383cc766b548e8c9ecb02f0ac18c","family_id":"qf-density16-det-scaled-matrix","seed":202610164027,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,2],[3,4]],\"t\":0}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], t=0. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-2, [[0, 0], [0, 0]], 0, 0)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2, [[0, 0], [0, 0]], 0, 0)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0028","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ac9930ae7bf3d672fb186f087052532bb335d62dee446d0d5565b3b3a9456b4","family_id":"qf-density16-det-scaled-matrix","seed":202610164028,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,0],[0,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(1, [[-2, 0], [0, -2]], 4, 4)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[-2, 0], [0, -2]], 4, 4)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0029","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b3b27b91335a01afcf8b20503fcc136f565c1215caf5ecb84d54377d6ec14bf","family_id":"qf-density16-det-scaled-matrix","seed":202610164029,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,0],[0,1]],\"t\":9007199254740991}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], t=9007199254740991. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0030","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6633a180988dcdb4ca13363882a7dad694b80ce9f234da0879f7221d6cdb3614","family_id":"qf-density16-det-scaled-matrix","seed":202610164030,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-8,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-10, [[-8, 1], [2, 1]], -10, -10)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-10, [[-8, 1], [2, 1]], -10, -10)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0031","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97a1ff9ad5e94f2818cc2a60cb399f4e29c8e64fd7a62c3612ed8b7ca8aa23cb","family_id":"qf-density16-det-scaled-matrix","seed":202610164031,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-7,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-9, [[14, -2], [-4, -2]], -36, -36)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-9, [[14, -2], [-4, -2]], -36, -36)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0032","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc3514d2d3d240664208f68160bc69b69cb5d38b86ede453fbdfd38cbd51fc9a","family_id":"qf-density16-det-scaled-matrix","seed":202610164032,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-6,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-8, [[-6, 1], [2, 1]], -8, -8)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-8, [[-6, 1], [2, 1]], -8, -8)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0033","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1ecb8263a07c63f334e72fb199901d038fa20808c5e8c4bd63f5d0418a821e0","family_id":"qf-density16-det-scaled-matrix","seed":202610164033,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-5,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-7, [[10, -2], [-4, -2]], -28, -28)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-7, [[10, -2], [-4, -2]], -28, -28)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0034","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"263bc4e5e363b2583ec011bc0c1598a538650b1258838ff42622608ad87716bc","family_id":"qf-density16-det-scaled-matrix","seed":202610164034,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-4,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-6, [[-4, 1], [2, 1]], -6, -6)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-6, [[-4, 1], [2, 1]], -6, -6)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0035","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fb82f4fa7826f41c3cad984e240d01d6828e1b57c9fe5af163af47e739e9f3e","family_id":"qf-density16-det-scaled-matrix","seed":202610164035,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-3,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-5, [[6, -2], [-4, -2]], -20, -20)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-5, [[6, -2], [-4, -2]], -20, -20)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0036","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aff89ef5b42d1b35944979a991a901b9a347e181eee6b4ed34cfa45f12fa1ae","family_id":"qf-density16-det-scaled-matrix","seed":202610164036,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-2,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-4, [[-2, 1], [2, 1]], -4, -4)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-4, [[-2, 1], [2, 1]], -4, -4)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0037","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58bff2db6cdcb5d37013bf396a223b08573d686700534ced58111a862313824b","family_id":"qf-density16-det-scaled-matrix","seed":202610164037,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-1,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-3, [[2, -2], [-4, -2]], -12, -12)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3, [[2, -2], [-4, -2]], -12, -12)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0038","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8925f92d70141ce62928fada98f9c0c2f627840c4c4c58f5b2b18fcb91c38f5","family_id":"qf-density16-det-scaled-matrix","seed":202610164038,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[0,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-2, [[0, 1], [2, 1]], -2, -2)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2, [[0, 1], [2, 1]], -2, -2)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0039","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b427a27a2b6f4b78fd26dccefeb38efeb96fae8d9f2e03c06b82a02fce64aa42","family_id":"qf-density16-det-scaled-matrix","seed":202610164039,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(-1, [[-2, -2], [-4, -2]], -4, -4)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1, [[-2, -2], [-4, -2]], -4, -4)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0040","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84f44f2f6ab60d01c0ec98e11b43b2aafa428c682d82fbcbaf2866ee0e5a4594","family_id":"qf-density16-det-scaled-matrix","seed":202610164040,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[2,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(0, [[2, 1], [2, 1]], 0, 0)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [[2, 1], [2, 1]], 0, 0)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0041","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8c4349b2e89063224635bed961417f43863d513ece7ee4741e5aff61683c51a","family_id":"qf-density16-det-scaled-matrix","seed":202610164041,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[3,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(1, [[-6, -2], [-4, -2]], 4, 4)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[-6, -2], [-4, -2]], 4, 4)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0042","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5a98e8ea1e2deb89eedae2fb82d47d455c89759a75d0c6aeca81e6990488909","family_id":"qf-density16-det-scaled-matrix","seed":202610164042,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[4,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(2, [[4, 1], [2, 1]], 2, 2)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[4, 1], [2, 1]], 2, 2)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0043","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26ff8e592a8fe8f834d81eb2e64c4e0b5bbd3574f2ec25e3d41474956c5aa4a5","family_id":"qf-density16-det-scaled-matrix","seed":202610164043,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[5,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(3, [[-10, -2], [-4, -2]], 12, 12)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[-10, -2], [-4, -2]], 12, 12)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0044","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e42a48465e59d8b9a57d4020db89906f49c4e3c30a594ddbb946edb2ddb347a","family_id":"qf-density16-det-scaled-matrix","seed":202610164044,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[6,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(4, [[6, 1], [2, 1]], 4, 4)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[6, 1], [2, 1]], 4, 4)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0045","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"334005240593176c7099aadf896cf0bcc20b5a45ca37cda82274f726fcb78c54","family_id":"qf-density16-det-scaled-matrix","seed":202610164045,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[7,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(5, [[-14, -2], [-4, -2]], 20, 20)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[-14, -2], [-4, -2]], 20, 20)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0046","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66b961e29c0220ace63234ec134d64c16fa07d60ab9c1c7ac2b458a14488c677","family_id":"qf-density16-det-scaled-matrix","seed":202610164046,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[8,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(6, [[8, 1], [2, 1]], 6, 6)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[8, 1], [2, 1]], 6, 6)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0047","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29b2af9a7fefc6a63dced1c902c7ff34779772e31a35dc70fae000754eb9de54","family_id":"qf-density16-det-scaled-matrix","seed":202610164047,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[9,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(7, [[-18, -2], [-4, -2]], 28, 28)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [[-18, -2], [-4, -2]], 28, 28)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0048","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fe7789cba844bf1c80f80f0c0c86b273f9c3125f362cb65f8d37339d27fc0e5","family_id":"qf-density16-det-scaled-matrix","seed":202610164048,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[10,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(8, [[10, 1], [2, 1]], 8, 8)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [[10, 1], [2, 1]], 8, 8)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0049","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1501f664769d0949df0c4752ede8514b8804a45f262215ddf592bb922c8d4f0","family_id":"qf-density16-det-scaled-matrix","seed":202610164049,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[11,1],[2,1]],\"t\":-2}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(9, [[-22, -2], [-4, -2]], 36, 36)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [[-22, -2], [-4, -2]], 36, 36)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0050","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86b2854f4334312d7ce7a87c130919d5a2829769d37537e57581e4d4b6f5ea25","family_id":"qf-density16-det-scaled-matrix","seed":202610164050,"semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[12,1],[2,1]],\"t\":1}"},"fields":{"title":"Complete Matrix Scaling and Signed Determinant Routes","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.","answer":{"kind":"expression","value":"(10, [[12, 1], [2, 1]], 10, 10)","explanation":"Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [[12, 1], [2, 1]], 10, 10)."},"skills":["Complete Matrix Scaling and Signed Determinant Routes"],"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-20261012-advance-20261016-r04-determinant-0051","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d51c42e2d6f822f8dd00d43f76422d9c2648055b272c0df30423bd0173caa98","family_id":"qf-density16-det-product-two-orders","seed":202610164051,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,0],[0,1]],\"N\":[[1,0],[0,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], N=[[1, 0], [0, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, Yes)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, Yes)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0052","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a6f41d58dbe705b6fb13b3aecd7de7c314e532985aa20acda2c8598ca1f5c38","family_id":"qf-density16-det-product-two-orders","seed":202610164052,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,1],[0,1]],\"N\":[[1,0],[1,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [0, 1]], N=[[1, 0], [1, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0053","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9ca2defdc32a7bac37739f75f8b912f9e8150e168e7c59f2d2c167c7f3240b","family_id":"qf-density16-det-product-two-orders","seed":202610164053,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[0,0],[0,0]],\"N\":[[1,2],[3,4]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], N=[[1, 2], [3, 4]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, Yes)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, Yes)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0054","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"720c209882f4552ef976b575916539d0ac72b0f09eda1465f780df2fb4393faf","family_id":"qf-density16-det-product-two-orders","seed":202610164054,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-9,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0055","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"276c48056746f66176644db27540c2ec8f07bcc31189206d263c92eaf42545d0","family_id":"qf-density16-det-product-two-orders","seed":202610164055,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-8,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0056","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a3383cf6ab5b77e204063ec33bed036751c14bf4ab3a91a65fe67ebc7b5b68f","family_id":"qf-density16-det-product-two-orders","seed":202610164056,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-7,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0057","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3141dd597e87906d00e5e0ea4cd6082769cfb328c9efbd0d8148a4682355b44d","family_id":"qf-density16-det-product-two-orders","seed":202610164057,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-6,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0058","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47fc18ceba8cd6b148247b1c60146396fa3f5958c1bef1e9d2e18f9acea9e82e","family_id":"qf-density16-det-product-two-orders","seed":202610164058,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-5,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0059","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe7ebe9f60469c595555d399f816a4d16d1ff3ea18d0adb03962abecc80e8e1","family_id":"qf-density16-det-product-two-orders","seed":202610164059,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-4,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0060","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac6b1994b87ea35270cbfb576d4c5f2e867b8b80d211b67bd5b0434de5d73383","family_id":"qf-density16-det-product-two-orders","seed":202610164060,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-3,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0061","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c79f58aca9328d7edd22c535b2c7ad9d9ac6f5fd7a5a65e72c41a7c0d4d37d2","family_id":"qf-density16-det-product-two-orders","seed":202610164061,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-2,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0062","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae8cd56746e201706e075a3b347147fea42aae9beade26bad52ecb52fa116b33","family_id":"qf-density16-det-product-two-orders","seed":202610164062,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-1,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0063","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ea980f7096f73ae747ae7d59345e6dd9d253f3b2520ae75143c7f84e8584207","family_id":"qf-density16-det-product-two-orders","seed":202610164063,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[0,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0064","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc38a95117ca44feb9652d82ef6ec78989edb5f047aea8e7ff3d69f6b9ea4f07","family_id":"qf-density16-det-product-two-orders","seed":202610164064,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0065","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0843d7cc7cbc9d0676eacb1b586448f5409e5098012976e369e7ced6a6fe40fb","family_id":"qf-density16-det-product-two-orders","seed":202610164065,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[2,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0066","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c60f85025a3d2b44921338e1f9344b853c4375a25a5051df0293fc49eccfa3eb","family_id":"qf-density16-det-product-two-orders","seed":202610164066,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[3,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0067","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35290a7124f28c2d981cd4b601e96532e85764f1a94c55df669fd7099a310e81","family_id":"qf-density16-det-product-two-orders","seed":202610164067,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[4,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0068","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c6887f4b0d991b0664491be6738f153997c088795cf6bbfb128ac6078912eba","family_id":"qf-density16-det-product-two-orders","seed":202610164068,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[5,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0069","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03ea66177cf763c1741df05724b03285ff4d45ca22590f64c67c71f994222340","family_id":"qf-density16-det-product-two-orders","seed":202610164069,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[6,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0070","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28a8e9206daf53db0f6b0e08cbe07e6faf7f2feb86a222534f2130d33a8ec6b1","family_id":"qf-density16-det-product-two-orders","seed":202610164070,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[7,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0071","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cd1c62cc36f1323a323c9a7f0cae5f57f45d0582c0b58b9a7b8078cbbedee1a","family_id":"qf-density16-det-product-two-orders","seed":202610164071,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[8,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0072","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cb718fd5560347db10d10342325cae2a7e7eac8db49b58a409d047d1ca9eaa9","family_id":"qf-density16-det-product-two-orders","seed":202610164072,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[9,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0073","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a989efe211b2897ef93a4c3ae4294144b3ddeb33fa468b0141a8816fee71058","family_id":"qf-density16-det-product-two-orders","seed":202610164073,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[10,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0074","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08c6747ed1ff65d8252deac43060fe35c0ffafbba7699f017f20c3c0a5a9a47a","family_id":"qf-density16-det-product-two-orders","seed":202610164074,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[11,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0075","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ecec7eee732e8a23395acb4cb8e4a602f20897a4d0307a9f8eefd51f555217","family_id":"qf-density16-det-product-two-orders","seed":202610164075,"semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[12,1],[0,1]],\"N\":[[1,0],[2,1]]}"},"fields":{"title":"Both Matrix Product Orders, Determinants, and Equality","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).","answer":{"kind":"expression","value":"([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, No)","explanation":"Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, No)."},"skills":["Both Matrix Product Orders, Determinants, and Equality"],"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-20261012-advance-20261016-r04-determinant-0076","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8992365d0ad6e1f8597529a0a20609e9f8f07f3ab41885a6a2a00d2eb4c7b4a1","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164076,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[0,0],[0,0]],\"candidate\":0}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], 0, 0, 0, Yes, No)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], 0, 0, 0, Yes, No)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0077","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da17ef421365c4caf8a8edbb2a8b10dc80871aaacdf015b5904da630a678b21","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164077,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,2],[3,4]],\"candidate\":2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, 2, 4, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, 2, 4, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0078","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e247237a502d44756922aa9b9c8116ddd44ffefb2efaa9ab337cd06fcbca78c6","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164078,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,2],[3,4]],\"candidate\":-2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, 2, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, 2, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0079","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a1739417652d435d40d910f05ba7ddc69f68de9133a4b4a6813030a5fee7088","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164079,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-9,1],[2,1]],\"candidate\":-10}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [2, 1]], integer candidate=-10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-9, 1], [2, 1]], -11, 11, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-9, 1], [2, 1]], -11, 11, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0080","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f8aba3ca6769137543daed4da545d3ad25bf3a4f5d2908378c4850cabe9f9d","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164080,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-8,1],[2,1]],\"candidate\":-10}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]], integer candidate=-10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-8, 1], [2, 1]], -10, 10, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 1], [2, 1]], -10, 10, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0081","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a9ed2f471ff52320a9f7fad18b7ae5dce7a2b5461dc1282f64aaf9e8364239","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164081,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-7,1],[2,1]],\"candidate\":-8}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]], integer candidate=-8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], -9, 9, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], -9, 9, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0082","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c26620c3d09df8d1a804a39ee968f1c32bf0abb3600bb73856db746a27f4f3","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164082,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-6,1],[2,1]],\"candidate\":-8}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]], integer candidate=-8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], -8, 8, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], -8, 8, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0083","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7f4a156b3ece4964268bcb3ec75f702f73f6f66d9423072c5fe894ecb8dcfaf","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164083,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-5,1],[2,1]],\"candidate\":-6}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]], integer candidate=-6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], -7, 7, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], -7, 7, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0084","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f08de06f7b339428ed4544a07b671259995936e50a30bd621a7d359e896161ae","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164084,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-4,1],[2,1]],\"candidate\":-6}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]], integer candidate=-6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], -6, 6, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], -6, 6, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0085","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdd7f682533a019c9b121290625fd73f7bc7d325e1b342ba8a8bc003b0c1aca1","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164085,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-3,1],[2,1]],\"candidate\":-4}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]], integer candidate=-4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], -5, 5, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], -5, 5, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0086","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfc36c7a9c232c5140d98277338017dbc0efcae4f6cf6729c56ff770ccf9de81","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164086,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-2,1],[2,1]],\"candidate\":-4}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]], integer candidate=-4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], -4, 4, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], -4, 4, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0087","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12a08b5442d0eae4dbe0e95a94b5e2665953955f555b1401c98bc2020aa3fe59","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164087,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-1,1],[2,1]],\"candidate\":-2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], -3, 3, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], -3, 3, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0088","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb641dc48af0d871f8cfaf052396b628ac204e8e15b61f556c524377356a0e77","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164088,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[0,1],[2,1]],\"candidate\":-2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], -2, 2, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], -2, 2, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0089","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20ae5dafbc3ce2b863e2c72cf0baab2fc4a03c676c967731877cb3be27c5c9f","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164089,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,1],[2,1]],\"candidate\":0}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], -1, 1, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], -1, 1, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0090","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f9684ce607d1bdfbcdc6819aca5393244fd828d38f147f07a503b5b0f7ecd43","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164090,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[2,1],[2,1]],\"candidate\":0}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], 0, 0, 0, Yes, No)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], 0, 0, 0, Yes, No)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0091","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b61ad28b38c37b7c350126f318c739a0e704b8b6f70af6ddacbcdec36b94ca3","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164091,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[3,1],[2,1]],\"candidate\":2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], 1, 1, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], 1, 1, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0092","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e32cf786078342b648c55417dadfd078876744653f16f5bdf4adde1a030be70","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164092,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[4,1],[2,1]],\"candidate\":2}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0093","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"918ec59547da56f2c826684c2b52cf6750377aa8e6f2a8eab8f00990d51f7770","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164093,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[5,1],[2,1]],\"candidate\":4}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]], integer candidate=4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], 3, 3, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], 3, 3, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0094","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf1fe7b44e5fe3c2be48bb1394481aab119f4fc269d4e2afba29da07a0cd56f","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164094,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[6,1],[2,1]],\"candidate\":4}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]], integer candidate=4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], 4, 4, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], 4, 4, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0095","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25779ff367a5d65f848d7460f9a05f3db65af420724577998266d850602a8cce","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164095,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[7,1],[2,1]],\"candidate\":6}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]], integer candidate=6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], 5, 5, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], 5, 5, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0096","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ed8094c05df911c60f89ae3a2d6f7bcf8b2b3d39f1bb34246ae5a5e1018eac","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164096,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[8,1],[2,1]],\"candidate\":6}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]], integer candidate=6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], 6, 6, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], 6, 6, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0097","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d6e2885272c354e1270f6272896a7315cc6291cf57d168c997d90eb98ed5f9e","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164097,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[9,1],[2,1]],\"candidate\":8}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]], integer candidate=8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], 7, 7, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], 7, 7, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0098","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bfcb052fff539350c9bb6d689ad725bd094c29431b32b5650a92f38e971ab36","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164098,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[10,1],[2,1]],\"candidate\":8}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]], integer candidate=8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], 8, 8, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], 8, 8, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0099","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b503a121f12f32ee2cdc6de0b122f5bb6f8633dda48a1dcf16c458ac41f93db","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164099,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[11,1],[2,1]],\"candidate\":10}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]], integer candidate=10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], 9, 9, 1, No, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], 9, 9, 1, No, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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-20261012-advance-20261016-r04-determinant-0100","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aaf83384a22824cb34ba4fdb9c32ba3ace1577a386dcc5a81616b6564e01b59","family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164100,"semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[12,1],[2,1]],\"candidate\":10}"},"fields":{"title":"Signed Determinant versus Absolute-Value Candidate","stem":"Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]], integer candidate=10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], 10, 10, 0, Yes, Yes)","explanation":"Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], 10, 10, 0, Yes, Yes)."},"skills":["Signed Determinant versus Absolute-Value Candidate"],"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."}}