{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r01-matrix-0001","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b0f0c7b62e46ee7188361f0703bf04eadee76e57f2263f26c06285931bdddb","family_id":"qf-matrix-shape","seed":202610071001,"semantic_template":"qf-matrix-shape/{\"inputs\":[1,1],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(1, 1)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 1). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0002","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"307d9d1cd6e61c4b7d98ae9f523799e7dbaa0d3f7a9948b86994a9310c055bb6","family_id":"qf-matrix-shape","seed":202610071002,"semantic_template":"qf-matrix-shape/{\"inputs\":[1,2],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(1, 2)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 2). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0003","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7142bb1737b34ff50492334390042e21172d7048fc1d60aafa5a291a143a2260","family_id":"qf-matrix-shape","seed":202610071003,"semantic_template":"qf-matrix-shape/{\"inputs\":[1,3],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(1, 3)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 3). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0004","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0887a405f5bfbc1310fba9b33b2884dce117e7b7009f6662ba4b1e4a798e292","family_id":"qf-matrix-shape","seed":202610071004,"semantic_template":"qf-matrix-shape/{\"inputs\":[1,4],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(1, 4)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 4). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0005","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f215e22a9d31271f118ca4a007355bd2a2c06fa2e65a6aed9cc195cad4cfcae","family_id":"qf-matrix-shape","seed":202610071005,"semantic_template":"qf-matrix-shape/{\"inputs\":[1,5],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(1, 5)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 5). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0006","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1db2d63a62eda4aec6340a724922b20e507dce0a94d896cadfe61f6e8e44dc6","family_id":"qf-matrix-shape","seed":202610071006,"semantic_template":"qf-matrix-shape/{\"inputs\":[2,1],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-7], [-5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(2, 1)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 1). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0007","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcc6e7b402412d8fabc0607e8717bde665dbe2b6a0f2707dd8783120ef8ddd0c","family_id":"qf-matrix-shape","seed":202610071007,"semantic_template":"qf-matrix-shape/{\"inputs\":[2,2],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(2, 2)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 2). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0008","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f71caf5fec9e1ecd666b6947170899452e93d10911cbeb34949869dcbf75e169","family_id":"qf-matrix-shape","seed":202610071008,"semantic_template":"qf-matrix-shape/{\"inputs\":[2,3],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(2, 3)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 3). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0009","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ae163058637d4c8e2f7a8d2c2740aabda5b9acff9a4c8759911385789a823d6","family_id":"qf-matrix-shape","seed":202610071009,"semantic_template":"qf-matrix-shape/{\"inputs\":[2,4],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(2, 4)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 4). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0010","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"409419b324bc36b81773789614ed26ff9880128ae1a3e43d6c86df102260fb68","family_id":"qf-matrix-shape","seed":202610071010,"semantic_template":"qf-matrix-shape/{\"inputs\":[2,5],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(2, 5)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 5). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0011","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f20a2e79b14b8ea0fc68494942a733113a11aa6000b14350c22c3a5522b9915f","family_id":"qf-matrix-shape","seed":202610071011,"semantic_template":"qf-matrix-shape/{\"inputs\":[3,1],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(3, 1)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 1). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0012","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dda6fd8ba36b5df61a9a346fda9481e4b981f9fc78f4c8d95c16a38c115acafe","family_id":"qf-matrix-shape","seed":202610071012,"semantic_template":"qf-matrix-shape/{\"inputs\":[3,2],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(3, 2)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 2). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0013","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eff19a10e9ec6bf7f5512e95cabea538ce77fcac5bb17d6b463187e27ce2cc82","family_id":"qf-matrix-shape","seed":202610071013,"semantic_template":"qf-matrix-shape/{\"inputs\":[3,3],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(3, 3)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 3). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0014","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897612fb7692d375713832508d9ab20beb86a488b43bcf512e7b64dc034d1b57","family_id":"qf-matrix-shape","seed":202610071014,"semantic_template":"qf-matrix-shape/{\"inputs\":[3,4],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(3, 4)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 4). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0015","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e8b655be3542d0f04657ec027f6814a309a81d890b8eade4da725796770920","family_id":"qf-matrix-shape","seed":202610071015,"semantic_template":"qf-matrix-shape/{\"inputs\":[3,5],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(3, 5)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 5). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0016","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8be53e86e06af2672cbe46a357227083da57913c2d5792e1ee02117a7d42d5","family_id":"qf-matrix-shape","seed":202610071016,"semantic_template":"qf-matrix-shape/{\"inputs\":[4,1],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(4, 1)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 1). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0017","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3f063b4cfb8bc7089c950c7579e19e8ac0e8df90abf36e80d5359a60124aba2","family_id":"qf-matrix-shape","seed":202610071017,"semantic_template":"qf-matrix-shape/{\"inputs\":[4,2],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(4, 2)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 2). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0018","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ddf57165bef1479594b466d3a68a135d46eb1ce797e95442793b559a354d79","family_id":"qf-matrix-shape","seed":202610071018,"semantic_template":"qf-matrix-shape/{\"inputs\":[4,3],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(4, 3)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 3). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0019","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb13855285f32fe7e3cf7592bb3b7e5b263daeca52acfdecaa3935528fb2546a","family_id":"qf-matrix-shape","seed":202610071019,"semantic_template":"qf-matrix-shape/{\"inputs\":[4,4],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(4, 4)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 4). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0020","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a403d66cf77e34af7657792a70b2b2f88b8dc33134f7d193ee3c28ddffc79c6","family_id":"qf-matrix-shape","seed":202610071020,"semantic_template":"qf-matrix-shape/{\"inputs\":[4,5],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(4, 5)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 5). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0021","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"694c8cbd294e2f9f55cb4b6195ba93c9282620cd70e82e52ac5eee1a20df37ba","family_id":"qf-matrix-shape","seed":202610071021,"semantic_template":"qf-matrix-shape/{\"inputs\":[5,1],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(5, 1)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 1). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0022","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fd5e7fe2f0dbc0ab3410b6569ca57b05594d02e538b38ed6a251b574f025254","family_id":"qf-matrix-shape","seed":202610071022,"semantic_template":"qf-matrix-shape/{\"inputs\":[5,2],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(5, 2)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 2). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0023","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0578fd71a4b7326a85dd15154d3348a9e44b81b26a819a08f6f5d410cb7429cd","family_id":"qf-matrix-shape","seed":202610071023,"semantic_template":"qf-matrix-shape/{\"inputs\":[5,3],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(5, 3)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 3). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0024","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31398db9761b7b4f7fa392ce20ff0c84c951b14b02689284cdbdf6e4064c3a00","family_id":"qf-matrix-shape","seed":202610071024,"semantic_template":"qf-matrix-shape/{\"inputs\":[5,4],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(5, 4)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 4). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0025","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f33d7471eac3fd84f6a968dad7580df843da37c0f94e0b94600647d076a127","family_id":"qf-matrix-shape","seed":202610071025,"semantic_template":"qf-matrix-shape/{\"inputs\":[5,5],\"operation\":\"matrix_shape\"}"},"fields":{"title":"Dimensions of a Nonempty Matrix","stem":"The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).","answer":{"kind":"short_text","value":"(5, 5)","explanation":"Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 5). The total number of entries is not either dimension by itself."},"skills":["Dimensions of a Nonempty Matrix"],"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-20261006-advance-20261007-r01-matrix-0026","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e358b640735e0c41f21173579448d4c4788167e1ab63c4de77822b233f2551aa","family_id":"qf-matrix-entry","seed":202610071026,"semantic_template":"qf-matrix-entry/{\"inputs\":[1,1,1,1,\"-12\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"-12","explanation":"Read M(1,1)=-12 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0027","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232c235e26b15beee0c020fa2994a0844d3c731cce338e369dafc8c0f72ca5a9","family_id":"qf-matrix-entry","seed":202610071027,"semantic_template":"qf-matrix-entry/{\"inputs\":[1,2,1,1,\"-11\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"-11","explanation":"Read M(1,1)=-11 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0028","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28250723581c15ce5dcdeecc885fd579adbd3d00824e02c3c05afba2cfe083d","family_id":"qf-matrix-entry","seed":202610071028,"semantic_template":"qf-matrix-entry/{\"inputs\":[1,3,1,1,\"-10\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"-10","explanation":"Read M(1,1)=-10 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0029","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646be089b4975d8549a85a401b694ffa060043001e1eeb9aeda91ba725dc918e","family_id":"qf-matrix-entry","seed":202610071029,"semantic_template":"qf-matrix-entry/{\"inputs\":[1,4,1,1,\"-9\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"-9","explanation":"Read M(1,1)=-9 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0030","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e085f773311ea0f608c3c94c57f463935c9cf6a4ebf9ea321ee61f104c5b74bc","family_id":"qf-matrix-entry","seed":202610071030,"semantic_template":"qf-matrix-entry/{\"inputs\":[1,5,1,1,\"-8\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"-8","explanation":"Read M(1,1)=-8 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0031","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2b7f9f587d30811a495e2d81a090000447a2c8874d93fcc5f7591a821b46fbb","family_id":"qf-matrix-entry","seed":202610071031,"semantic_template":"qf-matrix-entry/{\"inputs\":[2,1,2,1,\"-5\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-7], [-5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.","answer":{"kind":"short_text","value":"-5","explanation":"Read M(2,1)=-5 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0032","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c0547cd9bcfc8f65252983d364467585156095bc0469ffb91acac90f9868564","family_id":"qf-matrix-entry","seed":202610071032,"semantic_template":"qf-matrix-entry/{\"inputs\":[2,2,1,2,\"-7\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.","answer":{"kind":"short_text","value":"-7","explanation":"Read M(1,2)=-7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0033","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3111aa23871be1149ffc10f2974d20d2b004207b6185f4736bdf63152e8eceed","family_id":"qf-matrix-entry","seed":202610071033,"semantic_template":"qf-matrix-entry/{\"inputs\":[2,3,2,2,\"-4\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 2.","answer":{"kind":"short_text","value":"-4","explanation":"Read M(2,2)=-4 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0034","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e0831b9d39cbc161afc7e7ca687d05ac2d5ac32275a469971e8fd062dbcfa19","family_id":"qf-matrix-entry","seed":202610071034,"semantic_template":"qf-matrix-entry/{\"inputs\":[2,4,1,2,\"-5\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.","answer":{"kind":"short_text","value":"-5","explanation":"Read M(1,2)=-5 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0035","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5116c38b3a64d2c337420833243845ea1ae26f9f6df5f323304e3eb52eccbc48","family_id":"qf-matrix-entry","seed":202610071035,"semantic_template":"qf-matrix-entry/{\"inputs\":[2,5,2,2,\"-2\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 2.","answer":{"kind":"short_text","value":"-2","explanation":"Read M(2,2)=-2 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0036","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22c8ff370de5648095d68887a1edd1fe8692752bd41eab73fd3b86bbb74ee2f6","family_id":"qf-matrix-entry","seed":202610071036,"semantic_template":"qf-matrix-entry/{\"inputs\":[3,1,2,1,\"0\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.","answer":{"kind":"short_text","value":"0","explanation":"Read M(2,1)=0 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0037","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c76896d8cb075b4641bffb9c3cb775f6e227be921be5978f99cb0815713e110","family_id":"qf-matrix-entry","seed":202610071037,"semantic_template":"qf-matrix-entry/{\"inputs\":[3,2,3,1,\"3\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 1.","answer":{"kind":"short_text","value":"3","explanation":"Read M(3,1)=3 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0038","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b50dd3239811e1459ee0ca758958e055f8903145fc8f2167370afad5c220db64","family_id":"qf-matrix-entry","seed":202610071038,"semantic_template":"qf-matrix-entry/{\"inputs\":[3,3,1,3,\"-2\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 3.","answer":{"kind":"short_text","value":"-2","explanation":"Read M(1,3)=-2 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0039","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fa615e75a6e2df5b0f3174a7591b3e4603ea6ce522bcb16f36d90c3d38f4798","family_id":"qf-matrix-entry","seed":202610071039,"semantic_template":"qf-matrix-entry/{\"inputs\":[3,4,2,3,\"1\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 3.","answer":{"kind":"short_text","value":"1","explanation":"Read M(2,3)=1 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0040","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da4956fe114ef2ee191bf1fb9982b0f2b6fa12435bba9ec3fd7ed97342d64553","family_id":"qf-matrix-entry","seed":202610071040,"semantic_template":"qf-matrix-entry/{\"inputs\":[3,5,3,3,\"4\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 3.","answer":{"kind":"short_text","value":"4","explanation":"Read M(3,3)=4 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0041","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6e52b4ba9f08f0f98fd838dd5e3e159282396706a411b24c42382035da5851b","family_id":"qf-matrix-entry","seed":202610071041,"semantic_template":"qf-matrix-entry/{\"inputs\":[4,1,4,1,\"9\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 1.","answer":{"kind":"short_text","value":"9","explanation":"Read M(4,1)=9 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0042","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37a2fdf36a337b7d1f164f71a5a769587801262e2be4745ac8f9d2be8617bf39","family_id":"qf-matrix-entry","seed":202610071042,"semantic_template":"qf-matrix-entry/{\"inputs\":[4,2,1,2,\"3\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.","answer":{"kind":"short_text","value":"3","explanation":"Read M(1,2)=3 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0043","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeb44760a03ffad487527f2303de423e5d57c5a4a1420b4ed21a1c84dc7874f2","family_id":"qf-matrix-entry","seed":202610071043,"semantic_template":"qf-matrix-entry/{\"inputs\":[4,3,2,1,\"7\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.","answer":{"kind":"short_text","value":"7","explanation":"Read M(2,1)=7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0044","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94453901de820cf694c96aa52601a7c930272c8a2f0b8c7e007b31f6d3757754","family_id":"qf-matrix-entry","seed":202610071044,"semantic_template":"qf-matrix-entry/{\"inputs\":[4,4,3,4,\"7\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 4.","answer":{"kind":"short_text","value":"7","explanation":"Read M(3,4)=7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0045","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d3992db616e15bb0a74391a56592c1e6e37ecafbffbd208488320bb612a2c9","family_id":"qf-matrix-entry","seed":202610071045,"semantic_template":"qf-matrix-entry/{\"inputs\":[4,5,4,4,\"10\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 4.","answer":{"kind":"short_text","value":"10","explanation":"Read M(4,4)=10 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0046","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e60686bf4a244591d46b534f8093c0bcaf33b107b94e32ae97da25df28de19f6","family_id":"qf-matrix-entry","seed":202610071046,"semantic_template":"qf-matrix-entry/{\"inputs\":[5,1,1,1,\"8\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.","answer":{"kind":"short_text","value":"8","explanation":"Read M(1,1)=8 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0047","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e5204cef6f8cb63d0c09e553dd4221ed30a5b7fc022aff588e569f144ca3fe3","family_id":"qf-matrix-entry","seed":202610071047,"semantic_template":"qf-matrix-entry/{\"inputs\":[5,2,2,1,\"11\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.","answer":{"kind":"short_text","value":"11","explanation":"Read M(2,1)=11 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0048","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf91999eca1ce766ccaf53eb189137bf4099619297eaa149ab7e24f3767af3d3","family_id":"qf-matrix-entry","seed":202610071048,"semantic_template":"qf-matrix-entry/{\"inputs\":[5,3,3,2,\"13\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 2.","answer":{"kind":"short_text","value":"13","explanation":"Read M(3,2)=13 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0049","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"613cb22ad51ec355072331c79e385dec21826a73c46dd519c5dd2c1f7a8bc449","family_id":"qf-matrix-entry","seed":202610071049,"semantic_template":"qf-matrix-entry/{\"inputs\":[5,4,4,1,\"17\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 1.","answer":{"kind":"short_text","value":"17","explanation":"Read M(4,1)=17 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0050","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"144f431b58b1d55e597b9ea1841393f487e0f21de421634dc011208baf37e214","family_id":"qf-matrix-entry","seed":202610071050,"semantic_template":"qf-matrix-entry/{\"inputs\":[5,5,5,5,\"16\"],\"operation\":\"matrix_entry\"}"},"fields":{"title":"Specified Matrix Entry with One-Based Indices","stem":"The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 5, column 5.","answer":{"kind":"short_text","value":"16","explanation":"Read M(5,5)=16 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value."},"skills":["Specified Matrix Entry with One-Based Indices"],"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-20261006-advance-20261007-r01-matrix-0051","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9a3f1e9e1302d8373303049c0678c58f6f65163c4a0dae45e823a9c793ca939","family_id":"qf-matrix-addition","seed":202610071051,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-12\"]],[[\"12\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-12]] and B=[[12]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0]]","explanation":"Add entries in corresponding positions to obtain [[0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0052","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a9caa8e22a477016e717c3d7510a270263bcb6242773ca0aa21a8e3dee66480","family_id":"qf-matrix-addition","seed":202610071052,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-11\",\"-12\"]],[[\"2\",\"4\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-11, -12]] and B=[[2, 4]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0053","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17d5b8eadeedbb1d403dd9850f128506d976cdf503346f3e72736aac5891e263","family_id":"qf-matrix-addition","seed":202610071053,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-10\",\"-11\",\"-12\"]],[[\"1\",\"3\",\"5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-10, -11, -12]] and B=[[1, 3, 5]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0054","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"097b4f28790edaf92e7b8ca724002dd63bf55cfc13bf8538dbe5959ec47d1d9d","family_id":"qf-matrix-addition","seed":202610071054,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-9\",\"-10\",\"-11\",\"-12\"]],[[\"9\",\"10\",\"11\",\"12\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-9, -10, -11, -12]] and B=[[9, 10, 11, 12]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0, 0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0055","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a3cbfcba7367eafcc2f1aecd58092f17e08c6fede8d1a581f291eb8d0cf8211","family_id":"qf-matrix-addition","seed":202610071055,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-1\",\"1\",\"3\",\"5\",\"7\"]],[[\"-8\",\"-9\",\"-10\",\"-11\",\"-12\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-8, -9, -10, -11, -12]] and B=[[-1, 1, 3, 5, 7]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6, -5]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0056","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c14e5d0f5c004bf6eeb3bca119a015dd12fadc2dd1e73c373149897fa6f90163","family_id":"qf-matrix-addition","seed":202610071056,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-2\"],[\"-1\"]],[[\"-7\"],[\"-5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-7], [-5]] and B=[[-2], [-1]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9], [-6]]","explanation":"Add entries in corresponding positions to obtain [[-9], [-6]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0057","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9127f962d0938c749a3c84848805dd8b6d1e130f1e2be473b627f466b5ea497","family_id":"qf-matrix-addition","seed":202610071057,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-6\",\"-7\"],[\"-4\",\"-5\"]],[[\"6\",\"7\"],[\"4\",\"5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-6, -7], [-4, -5]] and B=[[6, 7], [4, 5]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0], [0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0], [0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0058","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b558fedc0483346f44cc60dc6afb33bf96d9590e9ca94b6390d782575d07970","family_id":"qf-matrix-addition","seed":202610071058,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-4\",\"-2\",\"0\"],[\"-3\",\"-1\",\"1\"]],[[\"-5\",\"-6\",\"-7\"],[\"-3\",\"-4\",\"-5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-5, -6, -7], [-3, -4, -5]] and B=[[-4, -2, 0], [-3, -1, 1]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7], [-6, -5, -4]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0059","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32cfa0a2b4bd3ebc332c8e81732f3f7b470df06e416512da083df77bc34ebe2f","family_id":"qf-matrix-addition","seed":202610071059,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-4\",\"-5\",\"-6\",\"-7\"],[\"-2\",\"-3\",\"-4\",\"-5\"]],[[\"-5\",\"-3\",\"-1\",\"1\"],[\"-4\",\"-2\",\"0\",\"2\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-4, -5, -6, -7], [-2, -3, -4, -5]] and B=[[-5, -3, -1, 1], [-4, -2, 0, 2]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6], [-6, -5, -4, -3]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0060","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"601253a5dfce3cc555ea6afe73a93c3f8721f06e458797466517450503c25398","family_id":"qf-matrix-addition","seed":202610071060,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-3\",\"-4\",\"-5\",\"-6\",\"-7\"],[\"-1\",\"-2\",\"-3\",\"-4\",\"-5\"]],[[\"3\",\"4\",\"5\",\"6\",\"7\"],[\"1\",\"2\",\"3\",\"4\",\"5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] and B=[[3, 4, 5, 6, 7], [1, 2, 3, 4, 5]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0061","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14337695687cfb6e908fa8c6ad5ce05a4d424aa3b28354aaf1afade1064e6df1","family_id":"qf-matrix-addition","seed":202610071061,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-2\"],[\"0\"],[\"2\"]],[[\"-7\"],[\"-6\"],[\"-5\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-2], [0], [2]] and B=[[-7], [-6], [-5]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9], [-6], [-3]]","explanation":"Add entries in corresponding positions to obtain [[-9], [-6], [-3]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0062","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4537f9aacf74c70799605d0ef08e84acf47b5ee561a051e666b593d8681c144d","family_id":"qf-matrix-addition","seed":202610071062,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-1\",\"-2\"],[\"1\",\"0\"],[\"3\",\"2\"]],[[\"-8\",\"-6\"],[\"-7\",\"-5\"],[\"-6\",\"-4\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[-1, -2], [1, 0], [3, 2]] and B=[[-8, -6], [-7, -5], [-6, -4]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8], [-6, -5], [-3, -2]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8], [-6, -5], [-3, -2]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0063","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18985a70f1ba0f36c991d58cd24702beec60b4abe384b5ae39f58eb94049dce8","family_id":"qf-matrix-addition","seed":202610071063,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"0\",\"-1\",\"-2\"],[\"2\",\"1\",\"0\"],[\"4\",\"3\",\"2\"]],[[\"0\",\"1\",\"2\"],[\"-2\",\"-1\",\"0\"],[\"-4\",\"-3\",\"-2\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] and B=[[0, 1, 2], [-2, -1, 0], [-4, -3, -2]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0, 0], [0, 0, 0], [0, 0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0, 0], [0, 0, 0], [0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0064","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b18243cf1e3ea14b7d68d888c83aa74d2f1b6f00abad24daf83ef6b825fb2bd","family_id":"qf-matrix-addition","seed":202610071064,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-10\",\"-8\",\"-6\",\"-4\"],[\"-9\",\"-7\",\"-5\",\"-3\"],[\"-8\",\"-6\",\"-4\",\"-2\"]],[[\"1\",\"0\",\"-1\",\"-2\"],[\"3\",\"2\",\"1\",\"0\"],[\"5\",\"4\",\"3\",\"2\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] and B=[[-10, -8, -6, -4], [-9, -7, -5, -3], [-8, -6, -4, -2]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0065","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2cd4ca456a792fe63e4d2fb0aacdfed387e40ff18098e0e361188ba19d49489","family_id":"qf-matrix-addition","seed":202610071065,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-11\",\"-9\",\"-7\",\"-5\",\"-3\"],[\"-10\",\"-8\",\"-6\",\"-4\",\"-2\"],[\"-9\",\"-7\",\"-5\",\"-3\",\"-1\"]],[[\"2\",\"1\",\"0\",\"-1\",\"-2\"],[\"4\",\"3\",\"2\",\"1\",\"0\"],[\"6\",\"5\",\"4\",\"3\",\"2\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] and B=[[-11, -9, -7, -5, -3], [-10, -8, -6, -4, -2], [-9, -7, -5, -3, -1]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0066","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4912a1ab22a0f72b65287f8d1723d8cc7761e8c7d49fdeebc41815799160283a","family_id":"qf-matrix-addition","seed":202610071066,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-3\"],[\"-5\"],[\"-7\"],[\"-9\"]],[[\"3\"],[\"5\"],[\"7\"],[\"9\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[3], [5], [7], [9]] and B=[[-3], [-5], [-7], [-9]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0], [0], [0], [0]]","explanation":"Add entries in corresponding positions to obtain [[0], [0], [0], [0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0067","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560817750b6b129169da8ed7b3dea6d7d63b517c2b9a8c55180a1e0329ab72b9","family_id":"qf-matrix-addition","seed":202610071067,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-13\",\"-11\"],[\"-12\",\"-10\"],[\"-11\",\"-9\"],[\"-10\",\"-8\"]],[[\"4\",\"3\"],[\"6\",\"5\"],[\"8\",\"7\"],[\"10\",\"9\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[4, 3], [6, 5], [8, 7], [10, 9]] and B=[[-13, -11], [-12, -10], [-11, -9], [-10, -8]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8], [-6, -5], [-3, -2], [0, 1]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8], [-6, -5], [-3, -2], [0, 1]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0068","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd1395387fd14947db562cedbb67fb93634b7fd5e9ecba6fa87a8cfbe4b29261","family_id":"qf-matrix-addition","seed":202610071068,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-14\",\"-12\",\"-10\"],[\"-13\",\"-11\",\"-9\"],[\"-12\",\"-10\",\"-8\"],[\"-11\",\"-9\",\"-7\"]],[[\"5\",\"4\",\"3\"],[\"7\",\"6\",\"5\"],[\"9\",\"8\",\"7\"],[\"11\",\"10\",\"9\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] and B=[[-14, -12, -10], [-13, -11, -9], [-12, -10, -8], [-11, -9, -7]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0069","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ece5814775c5e7e48e0f27e5158895d5b8c27c9975a9ef6c2d3040f80a6442f","family_id":"qf-matrix-addition","seed":202610071069,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-6\",\"-5\",\"-4\",\"-3\"],[\"-8\",\"-7\",\"-6\",\"-5\"],[\"-10\",\"-9\",\"-8\",\"-7\"],[\"-12\",\"-11\",\"-10\",\"-9\"]],[[\"6\",\"5\",\"4\",\"3\"],[\"8\",\"7\",\"6\",\"5\"],[\"10\",\"9\",\"8\",\"7\"],[\"12\",\"11\",\"10\",\"9\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] and B=[[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0070","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca7ac170eef90af54de7cf213c9da22b62a0d060def0c2d12513b73044556c3","family_id":"qf-matrix-addition","seed":202610071070,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-16\",\"-14\",\"-12\",\"-10\",\"-8\"],[\"-15\",\"-13\",\"-11\",\"-9\",\"-7\"],[\"-14\",\"-12\",\"-10\",\"-8\",\"-6\"],[\"-13\",\"-11\",\"-9\",\"-7\",\"-5\"]],[[\"7\",\"6\",\"5\",\"4\",\"3\"],[\"9\",\"8\",\"7\",\"6\",\"5\"],[\"11\",\"10\",\"9\",\"8\",\"7\"],[\"13\",\"12\",\"11\",\"10\",\"9\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] and B=[[-16, -14, -12, -10, -8], [-15, -13, -11, -9, -7], [-14, -12, -10, -8, -6], [-13, -11, -9, -7, -5]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1], [0, 1, 2, 3, 4]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1], [0, 1, 2, 3, 4]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0071","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"babb1c7aa081c65c26b7fa5f313dbed79e6d9152b4dd59ea95548bd30f70486c","family_id":"qf-matrix-addition","seed":202610071071,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-17\"],[\"-16\"],[\"-15\"],[\"-14\"],[\"-13\"]],[[\"8\"],[\"10\"],[\"12\"],[\"14\"],[\"16\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[8], [10], [12], [14], [16]] and B=[[-17], [-16], [-15], [-14], [-13]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9], [-6], [-3], [0], [3]]","explanation":"Add entries in corresponding positions to obtain [[-9], [-6], [-3], [0], [3]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0072","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63911d310f3bc53621ec6712a1bd1418ff42821da8731660be20dd67c6eb1eaf","family_id":"qf-matrix-addition","seed":202610071072,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-9\",\"-8\"],[\"-11\",\"-10\"],[\"-13\",\"-12\"],[\"-15\",\"-14\"],[\"-17\",\"-16\"]],[[\"9\",\"8\"],[\"11\",\"10\"],[\"13\",\"12\"],[\"15\",\"14\"],[\"17\",\"16\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] and B=[[-9, -8], [-11, -10], [-13, -12], [-15, -14], [-17, -16]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0073","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f61c0073f4fa049d59a9de2d5cbd376ce8a2ada69429a951fd44a2bf49f4cff0","family_id":"qf-matrix-addition","seed":202610071073,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-19\",\"-17\",\"-15\"],[\"-18\",\"-16\",\"-14\"],[\"-17\",\"-15\",\"-13\"],[\"-16\",\"-14\",\"-12\"],[\"-15\",\"-13\",\"-11\"]],[[\"10\",\"9\",\"8\"],[\"12\",\"11\",\"10\"],[\"14\",\"13\",\"12\"],[\"16\",\"15\",\"14\"],[\"18\",\"17\",\"16\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] and B=[[-19, -17, -15], [-18, -16, -14], [-17, -15, -13], [-16, -14, -12], [-15, -13, -11]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0074","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cc4b11ea03aa494611d614dca19312f64634ead597a70b6344a2cb5b06cb9bb","family_id":"qf-matrix-addition","seed":202610071074,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-20\",\"-18\",\"-16\",\"-14\"],[\"-19\",\"-17\",\"-15\",\"-13\"],[\"-18\",\"-16\",\"-14\",\"-12\"],[\"-17\",\"-15\",\"-13\",\"-11\"],[\"-16\",\"-14\",\"-12\",\"-10\"]],[[\"11\",\"10\",\"9\",\"8\"],[\"13\",\"12\",\"11\",\"10\"],[\"15\",\"14\",\"13\",\"12\"],[\"17\",\"16\",\"15\",\"14\"],[\"19\",\"18\",\"17\",\"16\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] and B=[[-20, -18, -16, -14], [-19, -17, -15, -13], [-18, -16, -14, -12], [-17, -15, -13, -11], [-16, -14, -12, -10]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0], [0, 1, 2, 3], [3, 4, 5, 6]]","explanation":"Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0], [0, 1, 2, 3], [3, 4, 5, 6]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0075","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31663b645913086eb683cf1300bd820d5e6c05c6a6e4ee75a078578e5c1a2f19","family_id":"qf-matrix-addition","seed":202610071075,"semantic_template":"qf-matrix-addition/{\"inputs\":[[[\"-12\",\"-11\",\"-10\",\"-9\",\"-8\"],[\"-14\",\"-13\",\"-12\",\"-11\",\"-10\"],[\"-16\",\"-15\",\"-14\",\"-13\",\"-12\"],[\"-18\",\"-17\",\"-16\",\"-15\",\"-14\"],[\"-20\",\"-19\",\"-18\",\"-17\",\"-16\"]],[[\"12\",\"11\",\"10\",\"9\",\"8\"],[\"14\",\"13\",\"12\",\"11\",\"10\"],[\"16\",\"15\",\"14\",\"13\",\"12\"],[\"18\",\"17\",\"16\",\"15\",\"14\"],[\"20\",\"19\",\"18\",\"17\",\"16\"]]],\"operation\":\"matrix_addition\"}"},"fields":{"title":"Addition of Equal-Size Matrices","stem":"Over R, A=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] and B=[[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]] have the same dimensions. Find A+B as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]","explanation":"Add entries in corresponding positions to obtain [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation."},"skills":["Addition of Equal-Size Matrices"],"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-20261006-advance-20261007-r01-matrix-0076","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1650a188514df963c05f3b0fd1d8e7931b03a8761ba2994113ed213c48ed60c8","family_id":"qf-matrix-transpose","seed":202610071076,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-12\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-12]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-12]]. The dimensions change from 1×1 to 1×1; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0077","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4240edf211c82d930a5c16aa53c447dacba9fdf24d1f88cac179f21a331bd926","family_id":"qf-matrix-transpose","seed":202610071077,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-11\",\"-12\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-11], [-12]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-11], [-12]]. The dimensions change from 1×2 to 2×1; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0078","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60de8af9e95dd0313a995efe30917ec0c8874fda7ff5e1020b9656886a815c1c","family_id":"qf-matrix-transpose","seed":202610071078,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-10\",\"-11\",\"-12\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-10], [-11], [-12]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-10], [-11], [-12]]. The dimensions change from 1×3 to 3×1; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0079","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c30340cc264c6e181a530f9961a77849934287b3ac6ea47c0c054f73da6b3cdc","family_id":"qf-matrix-transpose","seed":202610071079,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-9\",\"-10\",\"-11\",\"-12\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-9], [-10], [-11], [-12]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-9], [-10], [-11], [-12]]. The dimensions change from 1×4 to 4×1; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0080","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8793141de245ee5774573e7dc3bc2acc5b4bf2cd0e7cb82fd4cdb822ce77042","family_id":"qf-matrix-transpose","seed":202610071080,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-8\",\"-9\",\"-10\",\"-11\",\"-12\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-8], [-9], [-10], [-11], [-12]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-8], [-9], [-10], [-11], [-12]]. The dimensions change from 1×5 to 5×1; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0081","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdafd5a9a04231bd978b7fbb9098d68f4bcf4cb3ac32fc3c8295a0b0404d8df6","family_id":"qf-matrix-transpose","seed":202610071081,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-7\"],[\"-5\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-7], [-5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-7, -5]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-7, -5]]. The dimensions change from 2×1 to 1×2; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0082","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d63d05c8fadf33c995f7fc925dace807803737aff1440a16335205318840134a","family_id":"qf-matrix-transpose","seed":202610071082,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-6\",\"-7\"],[\"-4\",\"-5\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-6, -4], [-7, -5]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-6, -4], [-7, -5]]. The dimensions change from 2×2 to 2×2; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0083","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78b8d1ab0df8a3c2673303466d3ab1ccb89c701a34d48c7c18a7ab61505cf313","family_id":"qf-matrix-transpose","seed":202610071083,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-5\",\"-6\",\"-7\"],[\"-3\",\"-4\",\"-5\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-5, -3], [-6, -4], [-7, -5]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×3 to 3×2; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0084","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81bb492558f7ed6bd76ccbd490e6a3ac625e5ff27378fab5db4b7440f8c79ff1","family_id":"qf-matrix-transpose","seed":202610071084,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-4\",\"-5\",\"-6\",\"-7\"],[\"-2\",\"-3\",\"-4\",\"-5\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-4, -2], [-5, -3], [-6, -4], [-7, -5]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-4, -2], [-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×4 to 4×2; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0085","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a40c5e152cb669363977fca531ceb909adb4fa85527141f3b4eeb76d98de443","family_id":"qf-matrix-transpose","seed":202610071085,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-3\",\"-4\",\"-5\",\"-6\",\"-7\"],[\"-1\",\"-2\",\"-3\",\"-4\",\"-5\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-3, -1], [-4, -2], [-5, -3], [-6, -4], [-7, -5]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-3, -1], [-4, -2], [-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×5 to 5×2; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0086","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c78f08fe25e17efbbdd0183bc584be2203662f4651d4e8a20d1b7f8202f107ec","family_id":"qf-matrix-transpose","seed":202610071086,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-2\"],[\"0\"],[\"2\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-2, 0, 2]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-2, 0, 2]]. The dimensions change from 3×1 to 1×3; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0087","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3adde2d1f33fd412bfaead0fc27a34446f2fad331abc571712c75c9abc121f","family_id":"qf-matrix-transpose","seed":202610071087,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"-1\",\"-2\"],[\"1\",\"0\"],[\"3\",\"2\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[-1, 1, 3], [-2, 0, 2]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×2 to 2×3; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0088","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"806f06d9b6236cba141176bac2286ea182789e705430e05bb2e7b5fa476fad75","family_id":"qf-matrix-transpose","seed":202610071088,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"0\",\"-1\",\"-2\"],[\"2\",\"1\",\"0\"],[\"4\",\"3\",\"2\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[0, 2, 4], [-1, 1, 3], [-2, 0, 2]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×3 to 3×3; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0089","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e339a31f1d714cc62fab00bb6c88015385f5d0237722612e35e377202b905b","family_id":"qf-matrix-transpose","seed":202610071089,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"1\",\"0\",\"-1\",\"-2\"],[\"3\",\"2\",\"1\",\"0\"],[\"5\",\"4\",\"3\",\"2\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×4 to 4×3; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0090","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0078caa7e6759576b95ba62f774e5ba1f123aa1d93ca0743b36b6995905b1197","family_id":"qf-matrix-transpose","seed":202610071090,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"2\",\"1\",\"0\",\"-1\",\"-2\"],[\"4\",\"3\",\"2\",\"1\",\"0\"],[\"6\",\"5\",\"4\",\"3\",\"2\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[2, 4, 6], [1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[2, 4, 6], [1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×5 to 5×3; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0091","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abcb5178306f80098a33f50cd16ab014ba353b55616a9f533042bf9eb5d84903","family_id":"qf-matrix-transpose","seed":202610071091,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"3\"],[\"5\"],[\"7\"],[\"9\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[3, 5, 7, 9]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[3, 5, 7, 9]]. The dimensions change from 4×1 to 1×4; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0092","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8111591f111efbf7e2315f3b635f6f9a0eeba78c1f62b424f6cf62054eb17dac","family_id":"qf-matrix-transpose","seed":202610071092,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"4\",\"3\"],[\"6\",\"5\"],[\"8\",\"7\"],[\"10\",\"9\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[4, 6, 8, 10], [3, 5, 7, 9]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×2 to 2×4; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0093","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13abe4ee5d9887e41ca8cfd1c7626f839bc507023fce9c6128141bd122e3a77e","family_id":"qf-matrix-transpose","seed":202610071093,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"5\",\"4\",\"3\"],[\"7\",\"6\",\"5\"],[\"9\",\"8\",\"7\"],[\"11\",\"10\",\"9\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×3 to 3×4; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0094","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7a60d722460a896a457db9ff97eb76f9ec329fa096357b62e8df501290ab9d6","family_id":"qf-matrix-transpose","seed":202610071094,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"6\",\"5\",\"4\",\"3\"],[\"8\",\"7\",\"6\",\"5\"],[\"10\",\"9\",\"8\",\"7\"],[\"12\",\"11\",\"10\",\"9\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×4 to 4×4; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0095","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf7bd9edfbfe391c6ccfffd81b71faec60bec0bd5635560f69b9b35907440858","family_id":"qf-matrix-transpose","seed":202610071095,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"7\",\"6\",\"5\",\"4\",\"3\"],[\"9\",\"8\",\"7\",\"6\",\"5\"],[\"11\",\"10\",\"9\",\"8\",\"7\"],[\"13\",\"12\",\"11\",\"10\",\"9\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[7, 9, 11, 13], [6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[7, 9, 11, 13], [6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×5 to 5×4; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0096","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfae484efa5232df3aa2064a6e7df4e1e5be5ce9829558208b009853ec288a29","family_id":"qf-matrix-transpose","seed":202610071096,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"8\"],[\"10\"],[\"12\"],[\"14\"],[\"16\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[8, 10, 12, 14, 16]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[8, 10, 12, 14, 16]]. The dimensions change from 5×1 to 1×5; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0097","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04cea39f6a037562bafd318088cf2f7e3cfcf5eeeaa3aa4e792940dd8b6a6947","family_id":"qf-matrix-transpose","seed":202610071097,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"9\",\"8\"],[\"11\",\"10\"],[\"13\",\"12\"],[\"15\",\"14\"],[\"17\",\"16\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×2 to 2×5; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0098","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dbe2e06ed8a27bcdde4d2433a7499dcff6bbd0ecb4c70b0507c22d38dc5fa93","family_id":"qf-matrix-transpose","seed":202610071098,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"10\",\"9\",\"8\"],[\"12\",\"11\",\"10\"],[\"14\",\"13\",\"12\"],[\"16\",\"15\",\"14\"],[\"18\",\"17\",\"16\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×3 to 3×5; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0099","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f2fdea5d819d8e03a82289b990e5b78ef30a3a67067e766ba53457f5976ac4","family_id":"qf-matrix-transpose","seed":202610071099,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"11\",\"10\",\"9\",\"8\"],[\"13\",\"12\",\"11\",\"10\"],[\"15\",\"14\",\"13\",\"12\"],[\"17\",\"16\",\"15\",\"14\"],[\"19\",\"18\",\"17\",\"16\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×4 to 4×5; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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-20261006-advance-20261007-r01-matrix-0100","batch_id":"batch-20261006-advance-20261007-r01-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96350993729c143dabf32e65b2e7cc05fa0b6f138f8fdc62fbf97d7853dad975","family_id":"qf-matrix-transpose","seed":202610071100,"semantic_template":"qf-matrix-transpose/{\"inputs\":[[\"12\",\"11\",\"10\",\"9\",\"8\"],[\"14\",\"13\",\"12\",\"11\",\"10\"],[\"16\",\"15\",\"14\",\"13\",\"12\"],[\"18\",\"17\",\"16\",\"15\",\"14\"],[\"20\",\"19\",\"18\",\"17\",\"16\"]],\"operation\":\"matrix_transpose\"}"},"fields":{"title":"Transpose of a Rectangular Matrix","stem":"The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.","answer":{"kind":"expression","value":"[[12, 14, 16, 18, 20], [11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]","explanation":"Move each entry from row i, column j to row j, column i, obtaining [[12, 14, 16, 18, 20], [11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×5 to 5×5; this is not row reversal or matrix inversion."},"skills":["Transpose of a Rectangular Matrix"],"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."}}