{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r03-determinant-0001","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6d25ee606a1c6d553253806c53caefe1a38e44dc685f064631c046cf71bbac3","family_id":"qf-determinant-two-by-two","seed":202610073001,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"0","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=0. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0002","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2b498e78f9f2e7ac6336e8db6f424d4c7cf9c045346868d4cc7cec57f6ab846","family_id":"qf-determinant-two-by-two","seed":202610073002,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"1\",\"2\"],[\"2\",\"4\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"0","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=0. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0003","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7761f51049986ea1ff237b32b5dcd8ac4ac50036e1c936f646ddb122d53667a0","family_id":"qf-determinant-two-by-two","seed":202610073003,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"1\"],[\"1\",\"0\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-1","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-1. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0004","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b7fd9d1797c17fe95117f98a2dc392de8bb10143a68626fcf48f25ed339edf0","family_id":"qf-determinant-two-by-two","seed":202610073004,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-28","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-28. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0005","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f333df9f6ad8fb6c596c5fb44c607d922b940e9914775816fe366240c197ab","family_id":"qf-determinant-two-by-two","seed":202610073005,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-38","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-38. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0006","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebb7e167116d6432762a35857dbe6315481ea487bbef194f1bf76813a097382","family_id":"qf-determinant-two-by-two","seed":202610073006,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-38","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-38. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0007","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80c925a8987c0b0d83cbf37aa1255d7be197e686c276907a8dc99b4fe7a62f1b","family_id":"qf-determinant-two-by-two","seed":202610073007,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-6\",\"3\"],[\"-2\",\"7\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-36","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-36. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0008","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c8fb8689f1828b11ebab3d5391f1236166a0c4e096ff105ef99d1fb6c882b21","family_id":"qf-determinant-two-by-two","seed":202610073008,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-5\",\"4\"],[\"-2\",\"8\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-32","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-32. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0009","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0658056f65d99f4088b59d4495ecc33d5d0d7968f9a4629ade20c69b856a23e8","family_id":"qf-determinant-two-by-two","seed":202610073009,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-34","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-34. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0010","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e7f0719ffa5d34af87192e860e4175291d4dbdd03de6f2985c1822b9764313","family_id":"qf-determinant-two-by-two","seed":202610073010,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-26","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-26. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0011","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8a1ac72038ee59f6cb3e19bcf1fe47712b12af827905147f2c202b0f3e3c7b","family_id":"qf-determinant-two-by-two","seed":202610073011,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-16","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-16. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0012","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92196e4e05abba330b98ea742b405c538952f1b453fe47f0ece608f7b4903ec9","family_id":"qf-determinant-two-by-two","seed":202610073012,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"-4","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-4. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0013","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"438657d85e5b7a2aa2601f32d97aa63c453441f3c4d9fe3d7e9e1fb263f82a25","family_id":"qf-determinant-two-by-two","seed":202610073013,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"1\"],[\"-2\",\"13\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"2","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=2. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0014","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf263b3f44f5e26ad0ffc7414a4215b0965206c24844757333d79844dfdb574","family_id":"qf-determinant-two-by-two","seed":202610073014,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"1\",\"2\"],[\"-2\",\"14\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"18","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=18. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0015","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5595a80f26f2dedb824226ceb4208e33ac64fa66e87af15abaf148772bfb6d3c","family_id":"qf-determinant-two-by-two","seed":202610073015,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"2\",\"3\"],[\"-2\",\"15\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"36","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=36. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0016","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe12dc2bf2b6cb57844251eabe49c2b19a440b973d3a514ac56f8332c48c325","family_id":"qf-determinant-two-by-two","seed":202610073016,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"3\",\"4\"],[\"-2\",\"16\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"56","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=56. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0017","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f604e897d39d3c31c3e560675c4d056d464833aaa0f3763cbd9276b0eee05d","family_id":"qf-determinant-two-by-two","seed":202610073017,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"4\",\"1\"],[\"-2\",\"17\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"70","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=70. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0018","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e648e557987d2253bf1adac3044cde151179e15b20ac008d4dd92b98a50bd372","family_id":"qf-determinant-two-by-two","seed":202610073018,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"5\",\"2\"],[\"-2\",\"18\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"94","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=94. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0019","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"503390945d2c08c172dbfc191f574e3ffa8d4c97935aa2145081bd1697f68e58","family_id":"qf-determinant-two-by-two","seed":202610073019,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"6\",\"3\"],[\"-2\",\"19\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"120","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=120. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0020","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1310a7009155eef461376cb72291434ad87b05351189f457312caca13f65c03","family_id":"qf-determinant-two-by-two","seed":202610073020,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"7\",\"4\"],[\"-2\",\"20\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"148","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=148. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0021","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d38ad84ee9701cfaeea373d007bba9f74aaf58ad6a8c56fd438a37c10f2006c8","family_id":"qf-determinant-two-by-two","seed":202610073021,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"8\",\"1\"],[\"-2\",\"21\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"170","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=170. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0022","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646dabeb6dfa0859c7751fe4f2d9d609d7d359fc1a672e7e4a03a123205cd1c7","family_id":"qf-determinant-two-by-two","seed":202610073022,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"9\",\"2\"],[\"-2\",\"22\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"202","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=202. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0023","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3daa657f1a0afe2fe318600221d646b9025e19656c41e359be3fa5498421eee3","family_id":"qf-determinant-two-by-two","seed":202610073023,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"10\",\"3\"],[\"-2\",\"23\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"236","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=236. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0024","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3331ee3359f02a8a8945db3bc98228219672830ba45a3da3d90ceb925566c628","family_id":"qf-determinant-two-by-two","seed":202610073024,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"11\",\"4\"],[\"-2\",\"24\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"272","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=272. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0025","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd799680093016ac4aa8d63d99b7e56dd3d75fa1ec5a3082dd09f646d138e25","family_id":"qf-determinant-two-by-two","seed":202610073025,"semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"12\",\"1\"],[\"-2\",\"25\"]],\"operation\":\"det2\"}"},"fields":{"title":"Exact 2×2 Determinant","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. Find det(M) exactly.","answer":{"kind":"short_text","value":"302","explanation":"Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=302. Negative, zero, and positive determinants are all valid outputs."},"skills":["Exact 2×2 Determinant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0026","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e32e7a1cddc7273447e0cfbed1dbe513ed97a3b210d10ffea7b73e46c4cc929","family_id":"qf-determinant-invertibility","seed":202610073026,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"No","explanation":"The determinant is 0. Over R, invertibility is equivalent to a nonzero determinant, giving No. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0027","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"651f019e48e9368eb6d17328e10f42026508ddb13680d94df33558b3a9c41a75","family_id":"qf-determinant-invertibility","seed":202610073027,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"1\",\"2\"],[\"2\",\"4\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"No","explanation":"The determinant is 0. Over R, invertibility is equivalent to a nonzero determinant, giving No. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0028","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a1ddcaeac429d0b3560f2993d1f2f6dcae09248ce8b278e55080d02727ea8db","family_id":"qf-determinant-invertibility","seed":202610073028,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"0\",\"1\"],[\"1\",\"0\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -1. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0029","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beabaae0f65b3b15c5733e7cad7f93d2896b6dcb2eb663499dd9aaba77760e86","family_id":"qf-determinant-invertibility","seed":202610073029,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -28. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0030","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c87db01b688634d1ec9623875a08df913a2854bad00c9c0bb95407fa29cecdca","family_id":"qf-determinant-invertibility","seed":202610073030,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -38. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0031","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebfc4a9ee82501909d6d631330c1d22907fabf57aa9be229158b9f73b1069e80","family_id":"qf-determinant-invertibility","seed":202610073031,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -38. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0032","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a698d1294fe2df797cbeeb2da17b7ecb1cde8af2b3441a158d626e7b235cee90","family_id":"qf-determinant-invertibility","seed":202610073032,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-6\",\"3\"],[\"-2\",\"7\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -36. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0033","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cfdc4f0e3518ff3ce4d2db97e398b551d1a768607d20e0c3eed8fe70b94d228","family_id":"qf-determinant-invertibility","seed":202610073033,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-5\",\"4\"],[\"-2\",\"8\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -32. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0034","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2811ca5ae911440846ccd8a627f52fa6b49421376051770c25c214cac2d17517","family_id":"qf-determinant-invertibility","seed":202610073034,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -34. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0035","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6276fa9180e14f34bfa754379c97b2322ded84e215a7b6d04fd7b8bd47b3a43","family_id":"qf-determinant-invertibility","seed":202610073035,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -26. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0036","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31538ae82a13307b4d77d5d1b5f547beaa9cb810383a35e487c3a4c33c6da828","family_id":"qf-determinant-invertibility","seed":202610073036,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -16. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0037","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5347ddb31803fe710021341b4a9da000abf05d156cdc556564a303f2dfd79e4c","family_id":"qf-determinant-invertibility","seed":202610073037,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is -4. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0038","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9276a82576fb2cf8bc8406e765dbfacaa69e2d3ab62a235d76dedc7c79ad72c4","family_id":"qf-determinant-invertibility","seed":202610073038,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"0\",\"1\"],[\"-2\",\"13\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 2. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0039","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4637295891ee478663bd673d5953c23a4ddf9d88702e2c02a81f73cd5ec24679","family_id":"qf-determinant-invertibility","seed":202610073039,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"1\",\"2\"],[\"-2\",\"14\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 18. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0040","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e084a0a85b58ae7e8a487ba9bbdb2dda3d720f468b822a90a402861eba7824e1","family_id":"qf-determinant-invertibility","seed":202610073040,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"2\",\"3\"],[\"-2\",\"15\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 36. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0041","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07fa8b21362c5f2d24e6d99bad447f0a48fd7409d2ba576a383d5b30e22df8ed","family_id":"qf-determinant-invertibility","seed":202610073041,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"3\",\"4\"],[\"-2\",\"16\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 56. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0042","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd2457e39ed46b04af7b631be8861d7c85e28b94fa1eaebb7d7fcf6e515821c5","family_id":"qf-determinant-invertibility","seed":202610073042,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"4\",\"1\"],[\"-2\",\"17\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 70. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0043","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6559be80de93b7a311e45b5fb5f4ea40ac7ec71e376ec9b02abe7f2726b575a","family_id":"qf-determinant-invertibility","seed":202610073043,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"5\",\"2\"],[\"-2\",\"18\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 94. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0044","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bbd29480bc1ce07542f638f2b8ced97ef326b1457ca36a36bbdc2e33f48597e","family_id":"qf-determinant-invertibility","seed":202610073044,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"6\",\"3\"],[\"-2\",\"19\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 120. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0045","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"628a8f7e62681f91bac5b0571bb2bb7c988dca4c3a18a2d56985d2f9a48fc684","family_id":"qf-determinant-invertibility","seed":202610073045,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"7\",\"4\"],[\"-2\",\"20\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 148. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0046","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7097e21942309631765c502f220d3e343af5194ec87fa36707f46c4417739f1d","family_id":"qf-determinant-invertibility","seed":202610073046,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"8\",\"1\"],[\"-2\",\"21\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 170. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0047","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84bdff175aeed4a6e9a4df8b6cbc97078b13d31a7c589f4c668c3ed8255b84a8","family_id":"qf-determinant-invertibility","seed":202610073047,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"9\",\"2\"],[\"-2\",\"22\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 202. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0048","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08ee789da6b15e467978358bb9dd47d70b2ed1102a1c6e052d98dd04a6eefba6","family_id":"qf-determinant-invertibility","seed":202610073048,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"10\",\"3\"],[\"-2\",\"23\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 236. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0049","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0456874d151627ab65001c61c1491c4920353d1d831502366842f48a5934665","family_id":"qf-determinant-invertibility","seed":202610073049,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"11\",\"4\"],[\"-2\",\"24\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 272. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0050","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f67d0789c2ff57474c26d7399c0c3b0282ce4f8bb75ac5c6a612af769b8ff6a","family_id":"qf-determinant-invertibility","seed":202610073050,"semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"12\",\"1\"],[\"-2\",\"25\"]],\"operation\":\"invertible\"}"},"fields":{"title":"Invertibility over the Real Field","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. Is M invertible over the real field, without requiring an integer-entry inverse?","answer":{"kind":"short_text","value":"Yes","explanation":"The determinant is 302. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."},"skills":["Invertibility over the Real Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0051","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f00104550e29758a7deb9a59781d47f113c3a382d3edc24dd6434b0f4a7bb231","family_id":"qf-determinant-product","seed":202610073051,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"0\"]],[[\"1\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[0, 0], [0, 0]] and B=[[1, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"0","explanation":"The determinants of A and B are 0 and 2, so det(AB)=0; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0052","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7e6eedfc85151add423ef21ab3dc1cf6c043d75015cfa5e585a0e54de34fb97","family_id":"qf-determinant-product","seed":202610073052,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],[[\"2\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[1, 2], [2, 4]] and B=[[2, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"0","explanation":"The determinants of A and B are 0 and 5, so det(AB)=0; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0053","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e08040d1b07a97ffb05596c13093107eebcb9f8181bf9f5db6429b987840691","family_id":"qf-determinant-product","seed":202610073053,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[[\"3\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[0, 1], [1, 0]] and B=[[3, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-8","explanation":"The determinants of A and B are -1 and 8, so det(AB)=-8; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0054","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"431f5c7740ac7eed32c5f3f62cc5491962b5ec5b7bfdebfc1933d82c3cba1011","family_id":"qf-determinant-product","seed":202610073054,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[[\"4\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-9, 4], [-2, 4]] and B=[[4, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-224","explanation":"The determinants of A and B are -28 and 8, so det(AB)=-224; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0055","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e66ce90eb1fff23075357e207be8b16863fb6b2052274fe23e045f6b6032be49","family_id":"qf-determinant-product","seed":202610073055,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[[\"5\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-8, 1], [-2, 5]] and B=[[5, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-418","explanation":"The determinants of A and B are -38 and 11, so det(AB)=-418; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0056","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3efa7b067c772ae8ef15e808f190fa2b9cda5b83c71b238f772f2fcb593062ff","family_id":"qf-determinant-product","seed":202610073056,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[[\"6\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-7, 2], [-2, 6]] and B=[[6, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-532","explanation":"The determinants of A and B are -38 and 14, so det(AB)=-532; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0057","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77bc329aa15a5c8160e73612ddf943bdcf0c7d023b671a2515fcee5138b09f78","family_id":"qf-determinant-product","seed":202610073057,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[[\"7\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-6, 3], [-2, 7]] and B=[[7, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-504","explanation":"The determinants of A and B are -36 and 14, so det(AB)=-504; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0058","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5b48f9c69c056b50d05386478f1350b6ed8839cf03420f1961cf0a5b4fa40c7","family_id":"qf-determinant-product","seed":202610073058,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[[\"8\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-5, 4], [-2, 8]] and B=[[8, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-544","explanation":"The determinants of A and B are -32 and 17, so det(AB)=-544; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0059","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57537f24eaf676430990874c72f13d2b41b32f80ef5c967fb826f791d36fa5a9","family_id":"qf-determinant-product","seed":202610073059,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[[\"9\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-4, 1], [-2, 9]] and B=[[9, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-680","explanation":"The determinants of A and B are -34 and 20, so det(AB)=-680; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0060","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc4b36fe08c3588b803c6d62673f98b2070f9ee26f73451e26becd94310b880c","family_id":"qf-determinant-product","seed":202610073060,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[[\"10\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-3, 2], [-2, 10]] and B=[[10, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-520","explanation":"The determinants of A and B are -26 and 20, so det(AB)=-520; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0061","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53e430fedf304bfd775fc1935f270a265797b24a23c95a7819c30bb4fd1fa737","family_id":"qf-determinant-product","seed":202610073061,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[[\"11\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-2, 3], [-2, 11]] and B=[[11, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-368","explanation":"The determinants of A and B are -16 and 23, so det(AB)=-368; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0062","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cdeeb7bd1b0960be93b2e449de4033a2da0107bb567f4d3766630fc3b913a9c","family_id":"qf-determinant-product","seed":202610073062,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[[\"12\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[-1, 4], [-2, 12]] and B=[[12, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"-104","explanation":"The determinants of A and B are -4 and 26, so det(AB)=-104; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0063","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbe45cbc56dab89cd1d8bd3ef8f2719f525353d8246d16674f95d124fea9266c","family_id":"qf-determinant-product","seed":202610073063,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[[\"13\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[0, 1], [-2, 13]] and B=[[13, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"52","explanation":"The determinants of A and B are 2 and 26, so det(AB)=52; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0064","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e9917befb142ca53d8eee9add4882df9960c4d74e3ea75d2a2c09f233e25fb","family_id":"qf-determinant-product","seed":202610073064,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[[\"14\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[1, 2], [-2, 14]] and B=[[14, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"522","explanation":"The determinants of A and B are 18 and 29, so det(AB)=522; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0065","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba62300503ea3069155e65764a1ea2a6d2037ff16f38fe73c239730ba0ff907a","family_id":"qf-determinant-product","seed":202610073065,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"15\",\"-1\"],[\"2\",\"2\"]],[[\"2\",\"3\"],[\"-2\",\"15\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[2, 3], [-2, 15]] and B=[[15, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"1152","explanation":"The determinants of A and B are 36 and 32, so det(AB)=1152; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0066","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0cf20efa3def08ccefa229aff3af11c762c6de2fa8446fdcc0c05aba1ee58b3","family_id":"qf-determinant-product","seed":202610073066,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"16\",\"-1\"],[\"0\",\"2\"]],[[\"3\",\"4\"],[\"-2\",\"16\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[3, 4], [-2, 16]] and B=[[16, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"1792","explanation":"The determinants of A and B are 56 and 32, so det(AB)=1792; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0067","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78651bade52a5efd0e4c4e9d7153f1d37fc4b0436d231a2b0c850f767231e24b","family_id":"qf-determinant-product","seed":202610073067,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"17\",\"-1\"],[\"1\",\"2\"]],[[\"4\",\"1\"],[\"-2\",\"17\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[4, 1], [-2, 17]] and B=[[17, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"2450","explanation":"The determinants of A and B are 70 and 35, so det(AB)=2450; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0068","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c57facb3ede5c2db25b8376b9996432920f7464550829da5b529dfea978d9cc5","family_id":"qf-determinant-product","seed":202610073068,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"18\",\"-1\"],[\"2\",\"2\"]],[[\"5\",\"2\"],[\"-2\",\"18\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[5, 2], [-2, 18]] and B=[[18, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"3572","explanation":"The determinants of A and B are 94 and 38, so det(AB)=3572; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0069","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"772db98fe2e5244f042038f70a38608a0b7f8b35510304d5d162912bc1846dbd","family_id":"qf-determinant-product","seed":202610073069,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"19\",\"-1\"],[\"0\",\"2\"]],[[\"6\",\"3\"],[\"-2\",\"19\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[6, 3], [-2, 19]] and B=[[19, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"4560","explanation":"The determinants of A and B are 120 and 38, so det(AB)=4560; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0070","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66fdd10cbe9b9659036e5dddc516072f9fc4bf755d56bdae59a95cdc9fd8cf4","family_id":"qf-determinant-product","seed":202610073070,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"20\",\"-1\"],[\"1\",\"2\"]],[[\"7\",\"4\"],[\"-2\",\"20\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[7, 4], [-2, 20]] and B=[[20, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"6068","explanation":"The determinants of A and B are 148 and 41, so det(AB)=6068; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0071","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eedb7a64fe5fa7bb623f70802bade8a67b32462f09d985aab88813d1227cbad","family_id":"qf-determinant-product","seed":202610073071,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"21\",\"-1\"],[\"2\",\"2\"]],[[\"8\",\"1\"],[\"-2\",\"21\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[8, 1], [-2, 21]] and B=[[21, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"7480","explanation":"The determinants of A and B are 170 and 44, so det(AB)=7480; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0072","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78191225580869c35ea4d1440a1e04850ef00affa7b71bece12a4678ba440ab2","family_id":"qf-determinant-product","seed":202610073072,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"22\",\"-1\"],[\"0\",\"2\"]],[[\"9\",\"2\"],[\"-2\",\"22\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[9, 2], [-2, 22]] and B=[[22, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"8888","explanation":"The determinants of A and B are 202 and 44, so det(AB)=8888; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0073","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99803fe81bcf52f5614e42a6824c3401314b49869cf18de35624203e5b27cc5f","family_id":"qf-determinant-product","seed":202610073073,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[[\"23\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[10, 3], [-2, 23]] and B=[[23, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"11092","explanation":"The determinants of A and B are 236 and 47, so det(AB)=11092; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0074","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b542557227f8da7a033d0ca3662d2167c72e3661aa38de5ac33d2f4fad5c901","family_id":"qf-determinant-product","seed":202610073074,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[[\"24\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[11, 4], [-2, 24]] and B=[[24, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"13600","explanation":"The determinants of A and B are 272 and 50, so det(AB)=13600; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0075","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"616e22aaa5436363098d20b7e8a6bf52751ef8d74ae333cc0eae8769d980b0a2","family_id":"qf-determinant-product","seed":202610073075,"semantic_template":"qf-determinant-product/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[[\"25\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"},"fields":{"title":"Determinant of a Matrix Product","stem":"Over R, A=[[12, 1], [-2, 25]] and B=[[25, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.","answer":{"kind":"short_text","value":"15100","explanation":"The determinants of A and B are 302 and 50, so det(AB)=15100; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA."},"skills":["Determinant of a Matrix Product"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0076","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c46b38274ec1505da446748a4fa03811988c5d94e17fee63c2459de1e0d42e1b","family_id":"qf-determinant-scalar","seed":202610073076,"semantic_template":"qf-determinant-scalar/{\"inputs\":[\"zero_scalar\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. For c=0, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"0","explanation":"In dimension two, det(cM)=c²det(M), giving 0. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0077","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d7303044cf7a884db8b788e08b00859b343dd3292c8dd21c5e67e089509b687","family_id":"qf-determinant-scalar","seed":202610073077,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. For c=-1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"0","explanation":"In dimension two, det(cM)=c²det(M), giving 0. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0078","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62619c17f8cfef0f3d01c77c8d28239c0832543d519a8427c26b85f75e7c8240","family_id":"qf-determinant-scalar","seed":202610073078,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-9","explanation":"In dimension two, det(cM)=c²det(M), giving -9. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0079","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a15e411b18a4d922c27b5f87ae1f1df39a4cadef0e19201a06cffcae557a56f4","family_id":"qf-determinant-scalar","seed":202610073079,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-448","explanation":"In dimension two, det(cM)=c²det(M), giving -448. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0080","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1262a0f57174b40bf0f21c35a73265ddf2a86f99813fe0e9c28121020bcc02b","family_id":"qf-determinant-scalar","seed":202610073080,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-38","explanation":"In dimension two, det(cM)=c²det(M), giving -38. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0081","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a64a7cdb9ecdabf7184fb3a57b968118bff4b3f07849dc34312bc87ddd5b1b","family_id":"qf-determinant-scalar","seed":202610073081,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"2\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. For c=-2, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-152","explanation":"In dimension two, det(cM)=c²det(M), giving -152. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0082","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab95fc096909780646114eb2d03c558a3a9d8ce842025bb68f2be63317cf52d4","family_id":"qf-determinant-scalar","seed":202610073082,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-324","explanation":"In dimension two, det(cM)=c²det(M), giving -324. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0083","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b9b57949d680accb78dfd6e3715cb49b2144782bdaa45c3d9716e7fd04ec87f","family_id":"qf-determinant-scalar","seed":202610073083,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-512","explanation":"In dimension two, det(cM)=c²det(M), giving -512. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0084","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"160052aadbaf8abd6a979ad8d1e33c3fcf06f931f3b4eb660a83ff8e411ac02d","family_id":"qf-determinant-scalar","seed":202610073084,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-34","explanation":"In dimension two, det(cM)=c²det(M), giving -34. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0085","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d97065ae8ce6c956dd9730ed6513671f7671236d988ebe01402c2b4dbf99582","family_id":"qf-determinant-scalar","seed":202610073085,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"2\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. For c=-2, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-104","explanation":"In dimension two, det(cM)=c²det(M), giving -104. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0086","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"895c1f22eb79c24ec9fa38fdd14a6b01450ba4a36134d549ce0b2b322bd734d2","family_id":"qf-determinant-scalar","seed":202610073086,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-144","explanation":"In dimension two, det(cM)=c²det(M), giving -144. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0087","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9b5c09b614286c5f7266d1306d6a989762cce5cfcba76e62903b3fbd1701507","family_id":"qf-determinant-scalar","seed":202610073087,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"-64","explanation":"In dimension two, det(cM)=c²det(M), giving -64. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0088","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43048edb92334647091539938849e0b05f8b5fec586ba8503c19b1b363116705","family_id":"qf-determinant-scalar","seed":202610073088,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"2","explanation":"In dimension two, det(cM)=c²det(M), giving 2. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0089","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab86812a6ce9376d578c0a4a3044717a00b4ca3f94b5966a52f053e9e505515","family_id":"qf-determinant-scalar","seed":202610073089,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],\"2\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. For c=-2, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"72","explanation":"In dimension two, det(cM)=c²det(M), giving 72. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0090","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dd27117051df87461d7cb62305782efc615b69af7972db48aa045b2c2520d34","family_id":"qf-determinant-scalar","seed":202610073090,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"324","explanation":"In dimension two, det(cM)=c²det(M), giving 324. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0091","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e8f80f51f98863843eb32608a3022147b5f4081881e88847edfed0cd5a4d368","family_id":"qf-determinant-scalar","seed":202610073091,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"896","explanation":"In dimension two, det(cM)=c²det(M), giving 896. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0092","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cacd9494dc7d12151a3a5f33bf41949467ec05d1acdce2e1202281fd16dda193","family_id":"qf-determinant-scalar","seed":202610073092,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"70","explanation":"In dimension two, det(cM)=c²det(M), giving 70. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0093","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c09e915f9db137082563d168aed6648b7bf12ec7097abe79ca9d34d6a240a1de","family_id":"qf-determinant-scalar","seed":202610073093,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],\"2\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. For c=-2, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"376","explanation":"In dimension two, det(cM)=c²det(M), giving 376. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0094","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"242adb62c79c0816cdf6d7a11c27cf99dd690aab11274c76594ac70b5445d3ac","family_id":"qf-determinant-scalar","seed":202610073094,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"1080","explanation":"In dimension two, det(cM)=c²det(M), giving 1080. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0095","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b74a9f17d67bd4f149e3f92af2b4e163eb52c41f557371f08ac1a22a923c47a","family_id":"qf-determinant-scalar","seed":202610073095,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"2368","explanation":"In dimension two, det(cM)=c²det(M), giving 2368. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0096","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e7f232e2fb1224983129be17336bdce2947ab4423b4c09590c462779da58cda","family_id":"qf-determinant-scalar","seed":202610073096,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"170","explanation":"In dimension two, det(cM)=c²det(M), giving 170. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0097","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d90b83a4878879dec835048fce29640189730dc4b629fd3eb4581c2ba080ff2c","family_id":"qf-determinant-scalar","seed":202610073097,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],\"2\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. For c=-2, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"808","explanation":"In dimension two, det(cM)=c²det(M), giving 808. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0098","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095b3b7186680a64669eadea47c19c5268a90f078ffef17b8853819ab17e3709","family_id":"qf-determinant-scalar","seed":202610073098,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],\"3\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. For c=3, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"2124","explanation":"In dimension two, det(cM)=c²det(M), giving 2124. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0099","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7de81c456d4d43342941bcb8b77b251f9acc2f9b225154e1d8cacf65afac96","family_id":"qf-determinant-scalar","seed":202610073099,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],\"4\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. For c=-4, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"4352","explanation":"In dimension two, det(cM)=c²det(M), giving 4352. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r03-determinant-0100","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddab5a7e5bd772f6ee34c3df6311a8f835d5812dd336241e0ac9adffcf35eee1","family_id":"qf-determinant-scalar","seed":202610073100,"semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],\"1\"],\"operation\":\"scaled_determinant\"}"},"fields":{"title":"2×2 Determinant after Scalar Multiplication","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. For c=1, find det(cM), with every entry multiplied by c.","answer":{"kind":"short_text","value":"302","explanation":"In dimension two, det(cM)=c²det(M), giving 302. A negative c still acts through its square; c=0 gives determinant zero."},"skills":["2×2 Determinant after Scalar Multiplication"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}