{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0001","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 0], [0, 0]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0)","explanation_zh":"有限結構核對=([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164001,"parameters":{"M":[[0,0],[0,0]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[0,0],[0,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0002","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 2], [3, 4]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0)","explanation_zh":"有限結構核對=([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164002,"parameters":{"M":[[1,2],[3,4]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0003","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-10, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0)","explanation_zh":"有限結構核對=([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164003,"parameters":{"M":[[-10,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-10,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0004","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-9, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0)","explanation_zh":"有限結構核對=([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164004,"parameters":{"M":[[-9,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-9,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0005","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-8, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0)","explanation_zh":"有限結構核對=([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164005,"parameters":{"M":[[-8,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-8,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0006","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-7, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0)","explanation_zh":"有限結構核對=([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164006,"parameters":{"M":[[-7,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-7,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0007","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-6, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0)","explanation_zh":"有限結構核對=([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164007,"parameters":{"M":[[-6,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-6,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0008","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-5, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0)","explanation_zh":"有限結構核對=([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164008,"parameters":{"M":[[-5,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-5,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0009","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-4, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0)","explanation_zh":"有限結構核對=([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164009,"parameters":{"M":[[-4,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-4,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0010","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-3, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0)","explanation_zh":"有限結構核對=([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164010,"parameters":{"M":[[-3,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-3,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0011","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-2, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0)","explanation_zh":"有限結構核對=([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164011,"parameters":{"M":[[-2,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0012","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-1, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0)","explanation_zh":"有限結構核對=([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164012,"parameters":{"M":[[-1,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[-1,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0013","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)","explanation_zh":"有限結構核對=([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164013,"parameters":{"M":[[0,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[0,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0014","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0)","explanation_zh":"有限結構核對=([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164014,"parameters":{"M":[[1,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[1,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0015","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[2, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0)","explanation_zh":"有限結構核對=([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164015,"parameters":{"M":[[2,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0016","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[3, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0)","explanation_zh":"有限結構核對=([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164016,"parameters":{"M":[[3,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[3,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0017","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[4, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0)","explanation_zh":"有限結構核對=([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164017,"parameters":{"M":[[4,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[4,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0018","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[5, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0)","explanation_zh":"有限結構核對=([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164018,"parameters":{"M":[[5,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[5,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0019","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[6, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0)","explanation_zh":"有限結構核對=([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164019,"parameters":{"M":[[6,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[6,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0020","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[7, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0)","explanation_zh":"有限結構核對=([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164020,"parameters":{"M":[[7,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[7,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0021","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[8, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0)","explanation_zh":"有限結構核對=([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164021,"parameters":{"M":[[8,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[8,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0022","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[9, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0)","explanation_zh":"有限結構核對=([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164022,"parameters":{"M":[[9,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[9,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0023","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[10, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0)","explanation_zh":"有限結構核對=([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164023,"parameters":{"M":[[10,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[10,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0024","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[11, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0)","explanation_zh":"有限結構核對=([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164024,"parameters":{"M":[[11,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[11,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0025","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"交換兩行的有號行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[12, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)","explanation_zh":"有限結構核對=([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["交換兩行的有號行列式"]},"variant":{"family_id":"qf-density16-det-row-swap-signed","seed":202610164025,"parameters":{"M":[[12,1],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-row-swap-signed","semantic_template":"qf-density16-det-row-swap-signed/{\"M\":[[12,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0026","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 0], [0, 0]]、t=0。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(0, [[0, 0], [0, 0]], 0, 0)","explanation_zh":"有限結構核對=(0, [[0, 0], [0, 0]], 0, 0)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164026,"parameters":{"M":[[0,0],[0,0]],"t":0,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[0,0],[0,0]],\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0027","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 2], [3, 4]]、t=0。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-2, [[0, 0], [0, 0]], 0, 0)","explanation_zh":"有限結構核對=(-2, [[0, 0], [0, 0]], 0, 0)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164027,"parameters":{"M":[[1,2],[3,4]],"t":0,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,2],[3,4]],\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0028","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 0], [0, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(1, [[-2, 0], [0, -2]], 4, 4)","explanation_zh":"有限結構核對=(1, [[-2, 0], [0, -2]], 4, 4)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164028,"parameters":{"M":[[1,0],[0,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,0],[0,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0029","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 0], [0, 1]]、t=9007199254740991。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081)","explanation_zh":"有限結構核對=(1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164029,"parameters":{"M":[[1,0],[0,1]],"t":9007199254740991,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,0],[0,1]],\"t\":9007199254740991}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0030","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-8, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-10, [[-8, 1], [2, 1]], -10, -10)","explanation_zh":"有限結構核對=(-10, [[-8, 1], [2, 1]], -10, -10)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164030,"parameters":{"M":[[-8,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-8,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0031","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-7, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-9, [[14, -2], [-4, -2]], -36, -36)","explanation_zh":"有限結構核對=(-9, [[14, -2], [-4, -2]], -36, -36)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164031,"parameters":{"M":[[-7,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-7,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0032","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-6, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-8, [[-6, 1], [2, 1]], -8, -8)","explanation_zh":"有限結構核對=(-8, [[-6, 1], [2, 1]], -8, -8)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164032,"parameters":{"M":[[-6,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-6,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0033","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-5, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-7, [[10, -2], [-4, -2]], -28, -28)","explanation_zh":"有限結構核對=(-7, [[10, -2], [-4, -2]], -28, -28)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164033,"parameters":{"M":[[-5,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-5,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0034","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-4, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-6, [[-4, 1], [2, 1]], -6, -6)","explanation_zh":"有限結構核對=(-6, [[-4, 1], [2, 1]], -6, -6)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164034,"parameters":{"M":[[-4,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-4,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0035","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-3, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-5, [[6, -2], [-4, -2]], -20, -20)","explanation_zh":"有限結構核對=(-5, [[6, -2], [-4, -2]], -20, -20)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164035,"parameters":{"M":[[-3,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-3,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0036","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-2, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-4, [[-2, 1], [2, 1]], -4, -4)","explanation_zh":"有限結構核對=(-4, [[-2, 1], [2, 1]], -4, -4)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164036,"parameters":{"M":[[-2,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-2,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0037","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-1, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-3, [[2, -2], [-4, -2]], -12, -12)","explanation_zh":"有限結構核對=(-3, [[2, -2], [-4, -2]], -12, -12)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164037,"parameters":{"M":[[-1,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[-1,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0038","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-2, [[0, 1], [2, 1]], -2, -2)","explanation_zh":"有限結構核對=(-2, [[0, 1], [2, 1]], -2, -2)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164038,"parameters":{"M":[[0,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[0,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0039","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(-1, [[-2, -2], [-4, -2]], -4, -4)","explanation_zh":"有限結構核對=(-1, [[-2, -2], [-4, -2]], -4, -4)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164039,"parameters":{"M":[[1,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[1,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0040","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[2, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(0, [[2, 1], [2, 1]], 0, 0)","explanation_zh":"有限結構核對=(0, [[2, 1], [2, 1]], 0, 0)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164040,"parameters":{"M":[[2,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[2,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0041","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[3, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(1, [[-6, -2], [-4, -2]], 4, 4)","explanation_zh":"有限結構核對=(1, [[-6, -2], [-4, -2]], 4, 4)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164041,"parameters":{"M":[[3,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[3,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0042","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[4, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(2, [[4, 1], [2, 1]], 2, 2)","explanation_zh":"有限結構核對=(2, [[4, 1], [2, 1]], 2, 2)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164042,"parameters":{"M":[[4,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[4,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0043","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[5, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(3, [[-10, -2], [-4, -2]], 12, 12)","explanation_zh":"有限結構核對=(3, [[-10, -2], [-4, -2]], 12, 12)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164043,"parameters":{"M":[[5,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[5,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0044","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[6, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(4, [[6, 1], [2, 1]], 4, 4)","explanation_zh":"有限結構核對=(4, [[6, 1], [2, 1]], 4, 4)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164044,"parameters":{"M":[[6,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[6,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0045","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[7, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(5, [[-14, -2], [-4, -2]], 20, 20)","explanation_zh":"有限結構核對=(5, [[-14, -2], [-4, -2]], 20, 20)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164045,"parameters":{"M":[[7,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[7,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0046","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[8, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(6, [[8, 1], [2, 1]], 6, 6)","explanation_zh":"有限結構核對=(6, [[8, 1], [2, 1]], 6, 6)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164046,"parameters":{"M":[[8,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[8,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0047","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[9, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(7, [[-18, -2], [-4, -2]], 28, 28)","explanation_zh":"有限結構核對=(7, [[-18, -2], [-4, -2]], 28, 28)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164047,"parameters":{"M":[[9,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[9,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0048","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[10, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(8, [[10, 1], [2, 1]], 8, 8)","explanation_zh":"有限結構核對=(8, [[10, 1], [2, 1]], 8, 8)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164048,"parameters":{"M":[[10,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[10,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0049","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[11, 1], [2, 1]]、t=-2。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(9, [[-22, -2], [-4, -2]], 36, 36)","explanation_zh":"有限結構核對=(9, [[-22, -2], [-4, -2]], 36, 36)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164049,"parameters":{"M":[[11,1],[2,1]],"t":-2,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[11,1],[2,1]],\"t\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0050","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量矩陣與二階縮放值","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[12, 1], [2, 1]]、t=1。求 (D(M), tM, D(tM), t²D(M))。負t及零t都按明示數值計算。","answer":{"kind":"expression","value":"(10, [[12, 1], [2, 1]], 10, 10)","explanation_zh":"有限結構核對=(10, [[12, 1], [2, 1]], 10, 10)。逐格乘出tM，另一計算路徑以t平方乘原有號D取得二階縮放值。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量矩陣與二階縮放值"]},"variant":{"family_id":"qf-density16-det-scaled-matrix","seed":202610164050,"parameters":{"M":[[12,1],[2,1]],"t":1,"domain":"R_2x2_integer_slice","operation":"det-scaled-matrix","semantic_template":"qf-density16-det-scaled-matrix/{\"M\":[[12,1],[2,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0051","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 0], [0, 1]]、N=[[1, 0], [0, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, 是)","explanation_zh":"有限結構核對=([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, 是)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164051,"parameters":{"M":[[1,0],[0,1]],"N":[[1,0],[0,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,0],[0,1]],\"N\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0052","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 1], [0, 1]]、N=[[1, 0], [1, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, 否)","explanation_zh":"有限結構核對=([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164052,"parameters":{"M":[[1,1],[0,1]],"N":[[1,0],[1,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,1],[0,1]],\"N\":[[1,0],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0053","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 0], [0, 0]]、N=[[1, 2], [3, 4]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, 是)","explanation_zh":"有限結構核對=([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, 是)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164053,"parameters":{"M":[[0,0],[0,0]],"N":[[1,2],[3,4]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[0,0],[0,0]],\"N\":[[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0054","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-9, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, 否)","explanation_zh":"有限結構核對=([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164054,"parameters":{"M":[[-9,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-9,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0055","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-8, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, 否)","explanation_zh":"有限結構核對=([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164055,"parameters":{"M":[[-8,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-8,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0056","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-7, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, 否)","explanation_zh":"有限結構核對=([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164056,"parameters":{"M":[[-7,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-7,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0057","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-6, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, 否)","explanation_zh":"有限結構核對=([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164057,"parameters":{"M":[[-6,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-6,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0058","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-5, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, 否)","explanation_zh":"有限結構核對=([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164058,"parameters":{"M":[[-5,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-5,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0059","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-4, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, 否)","explanation_zh":"有限結構核對=([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164059,"parameters":{"M":[[-4,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-4,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0060","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-3, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, 否)","explanation_zh":"有限結構核對=([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164060,"parameters":{"M":[[-3,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-3,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0061","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-2, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, 否)","explanation_zh":"有限結構核對=([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164061,"parameters":{"M":[[-2,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-2,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0062","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-1, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, 否)","explanation_zh":"有限結構核對=([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164062,"parameters":{"M":[[-1,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[-1,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0063","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, 否)","explanation_zh":"有限結構核對=([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164063,"parameters":{"M":[[0,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[0,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0064","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, 否)","explanation_zh":"有限結構核對=([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164064,"parameters":{"M":[[1,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[1,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0065","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[2, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, 否)","explanation_zh":"有限結構核對=([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164065,"parameters":{"M":[[2,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[2,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0066","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[3, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, 否)","explanation_zh":"有限結構核對=([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164066,"parameters":{"M":[[3,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[3,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0067","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[4, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, 否)","explanation_zh":"有限結構核對=([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164067,"parameters":{"M":[[4,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[4,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0068","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[5, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, 否)","explanation_zh":"有限結構核對=([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164068,"parameters":{"M":[[5,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[5,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0069","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[6, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, 否)","explanation_zh":"有限結構核對=([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164069,"parameters":{"M":[[6,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[6,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0070","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[7, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, 否)","explanation_zh":"有限結構核對=([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164070,"parameters":{"M":[[7,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[7,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0071","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[8, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, 否)","explanation_zh":"有限結構核對=([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164071,"parameters":{"M":[[8,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[8,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0072","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[9, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, 否)","explanation_zh":"有限結構核對=([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164072,"parameters":{"M":[[9,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[9,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0073","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[10, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, 否)","explanation_zh":"有限結構核對=([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164073,"parameters":{"M":[[10,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[10,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0074","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[11, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, 否)","explanation_zh":"有限結構核對=([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164074,"parameters":{"M":[[11,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[11,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0075","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積兩序及行列式","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[12, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。","answer":{"kind":"expression","value":"([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, 否)","explanation_zh":"有限結構核對=([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積兩序及行列式"]},"variant":{"family_id":"qf-density16-det-product-two-orders","seed":202610164075,"parameters":{"M":[[12,1],[0,1]],"N":[[1,0],[2,1]],"domain":"R_2x2_integer_slice","operation":"det-product-two-orders","semantic_template":"qf-density16-det-product-two-orders/{\"M\":[[12,1],[0,1]],\"N\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0076","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 0], [0, 0]]，整數候選=0。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[0, 0], [0, 0]], 0, 0, 0, 是, 否)","explanation_zh":"有限結構核對=([[0, 0], [0, 0]], 0, 0, 0, 是, 否)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164076,"parameters":{"M":[[0,0],[0,0]],"candidate":0,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[0,0],[0,0]],\"candidate\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0077","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 2], [3, 4]]，整數候選=2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, 2, 4, 否, 是)","explanation_zh":"有限結構核對=([[1, 2], [3, 4]], -2, 2, 4, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164077,"parameters":{"M":[[1,2],[3,4]],"candidate":2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,2],[3,4]],\"candidate\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0078","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 2], [3, 4]]，整數候選=-2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[1, 2], [3, 4]], -2, 2, 0, 是, 是)","explanation_zh":"有限結構核對=([[1, 2], [3, 4]], -2, 2, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164078,"parameters":{"M":[[1,2],[3,4]],"candidate":-2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,2],[3,4]],\"candidate\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0079","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-9, 1], [2, 1]]，整數候選=-10。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-9, 1], [2, 1]], -11, 11, 1, 否, 是)","explanation_zh":"有限結構核對=([[-9, 1], [2, 1]], -11, 11, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164079,"parameters":{"M":[[-9,1],[2,1]],"candidate":-10,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-9,1],[2,1]],\"candidate\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0080","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-8, 1], [2, 1]]，整數候選=-10。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-8, 1], [2, 1]], -10, 10, 0, 是, 是)","explanation_zh":"有限結構核對=([[-8, 1], [2, 1]], -10, 10, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164080,"parameters":{"M":[[-8,1],[2,1]],"candidate":-10,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-8,1],[2,1]],\"candidate\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0081","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-7, 1], [2, 1]]，整數候選=-8。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-7, 1], [2, 1]], -9, 9, 1, 否, 是)","explanation_zh":"有限結構核對=([[-7, 1], [2, 1]], -9, 9, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164081,"parameters":{"M":[[-7,1],[2,1]],"candidate":-8,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-7,1],[2,1]],\"candidate\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0082","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-6, 1], [2, 1]]，整數候選=-8。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-6, 1], [2, 1]], -8, 8, 0, 是, 是)","explanation_zh":"有限結構核對=([[-6, 1], [2, 1]], -8, 8, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164082,"parameters":{"M":[[-6,1],[2,1]],"candidate":-8,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-6,1],[2,1]],\"candidate\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0083","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-5, 1], [2, 1]]，整數候選=-6。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-5, 1], [2, 1]], -7, 7, 1, 否, 是)","explanation_zh":"有限結構核對=([[-5, 1], [2, 1]], -7, 7, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164083,"parameters":{"M":[[-5,1],[2,1]],"candidate":-6,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-5,1],[2,1]],\"candidate\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0084","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-4, 1], [2, 1]]，整數候選=-6。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-4, 1], [2, 1]], -6, 6, 0, 是, 是)","explanation_zh":"有限結構核對=([[-4, 1], [2, 1]], -6, 6, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164084,"parameters":{"M":[[-4,1],[2,1]],"candidate":-6,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-4,1],[2,1]],\"candidate\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0085","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-3, 1], [2, 1]]，整數候選=-4。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-3, 1], [2, 1]], -5, 5, 1, 否, 是)","explanation_zh":"有限結構核對=([[-3, 1], [2, 1]], -5, 5, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164085,"parameters":{"M":[[-3,1],[2,1]],"candidate":-4,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-3,1],[2,1]],\"candidate\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0086","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-2, 1], [2, 1]]，整數候選=-4。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-2, 1], [2, 1]], -4, 4, 0, 是, 是)","explanation_zh":"有限結構核對=([[-2, 1], [2, 1]], -4, 4, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164086,"parameters":{"M":[[-2,1],[2,1]],"candidate":-4,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-2,1],[2,1]],\"candidate\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0087","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[-1, 1], [2, 1]]，整數候選=-2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[-1, 1], [2, 1]], -3, 3, 1, 否, 是)","explanation_zh":"有限結構核對=([[-1, 1], [2, 1]], -3, 3, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164087,"parameters":{"M":[[-1,1],[2,1]],"candidate":-2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[-1,1],[2,1]],\"candidate\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0088","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[0, 1], [2, 1]]，整數候選=-2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[0, 1], [2, 1]], -2, 2, 0, 是, 是)","explanation_zh":"有限結構核對=([[0, 1], [2, 1]], -2, 2, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164088,"parameters":{"M":[[0,1],[2,1]],"candidate":-2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[0,1],[2,1]],\"candidate\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0089","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[1, 1], [2, 1]]，整數候選=0。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[1, 1], [2, 1]], -1, 1, 1, 否, 是)","explanation_zh":"有限結構核對=([[1, 1], [2, 1]], -1, 1, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164089,"parameters":{"M":[[1,1],[2,1]],"candidate":0,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[1,1],[2,1]],\"candidate\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0090","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[2, 1], [2, 1]]，整數候選=0。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[2, 1], [2, 1]], 0, 0, 0, 是, 否)","explanation_zh":"有限結構核對=([[2, 1], [2, 1]], 0, 0, 0, 是, 否)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164090,"parameters":{"M":[[2,1],[2,1]],"candidate":0,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[2,1],[2,1]],\"candidate\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0091","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[3, 1], [2, 1]]，整數候選=2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[3, 1], [2, 1]], 1, 1, 1, 否, 是)","explanation_zh":"有限結構核對=([[3, 1], [2, 1]], 1, 1, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164091,"parameters":{"M":[[3,1],[2,1]],"candidate":2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[3,1],[2,1]],\"candidate\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0092","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[4, 1], [2, 1]]，整數候選=2。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[4, 1], [2, 1]], 2, 2, 0, 是, 是)","explanation_zh":"有限結構核對=([[4, 1], [2, 1]], 2, 2, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164092,"parameters":{"M":[[4,1],[2,1]],"candidate":2,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[4,1],[2,1]],\"candidate\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0093","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[5, 1], [2, 1]]，整數候選=4。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[5, 1], [2, 1]], 3, 3, 1, 否, 是)","explanation_zh":"有限結構核對=([[5, 1], [2, 1]], 3, 3, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164093,"parameters":{"M":[[5,1],[2,1]],"candidate":4,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[5,1],[2,1]],\"candidate\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0094","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[6, 1], [2, 1]]，整數候選=4。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[6, 1], [2, 1]], 4, 4, 0, 是, 是)","explanation_zh":"有限結構核對=([[6, 1], [2, 1]], 4, 4, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164094,"parameters":{"M":[[6,1],[2,1]],"candidate":4,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[6,1],[2,1]],\"candidate\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0095","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[7, 1], [2, 1]]，整數候選=6。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[7, 1], [2, 1]], 5, 5, 1, 否, 是)","explanation_zh":"有限結構核對=([[7, 1], [2, 1]], 5, 5, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164095,"parameters":{"M":[[7,1],[2,1]],"candidate":6,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[7,1],[2,1]],\"candidate\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0096","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[8, 1], [2, 1]]，整數候選=6。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[8, 1], [2, 1]], 6, 6, 0, 是, 是)","explanation_zh":"有限結構核對=([[8, 1], [2, 1]], 6, 6, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164096,"parameters":{"M":[[8,1],[2,1]],"candidate":6,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[8,1],[2,1]],\"candidate\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0097","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[9, 1], [2, 1]]，整數候選=8。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[9, 1], [2, 1]], 7, 7, 1, 否, 是)","explanation_zh":"有限結構核對=([[9, 1], [2, 1]], 7, 7, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164097,"parameters":{"M":[[9,1],[2,1]],"candidate":8,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[9,1],[2,1]],\"candidate\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0098","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[10, 1], [2, 1]]，整數候選=8。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[10, 1], [2, 1]], 8, 8, 0, 是, 是)","explanation_zh":"有限結構核對=([[10, 1], [2, 1]], 8, 8, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164098,"parameters":{"M":[[10,1],[2,1]],"candidate":8,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[10,1],[2,1]],\"candidate\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0099","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[11, 1], [2, 1]]，整數候選=10。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[11, 1], [2, 1]], 9, 9, 1, 否, 是)","explanation_zh":"有限結構核對=([[11, 1], [2, 1]], 9, 9, 1, 否, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164099,"parameters":{"M":[[11,1],[2,1]],"candidate":10,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[11,1],[2,1]],\"candidate\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r04-determinant-0100","batch_id":"batch-20261012-advance-20261016-r04-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"有號D與絕對值候選","stem_zh":"R上2×2矩陣，D為有號行列式。M=[[12, 1], [2, 1]]，整數候選=10。求 (M, 有號D, abs(D), 候選減有號D, 候選正確資格, 在R上可逆資格)。","answer":{"kind":"expression","value":"([[12, 1], [2, 1]], 10, 10, 0, 是, 是)","explanation_zh":"有限結構核對=([[12, 1], [2, 1]], 10, 10, 0, 是, 是)。先保留有號乘積差；候選核有號真值，R上的可逆條件另核D非零。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有號D與絕對值候選"]},"variant":{"family_id":"qf-density16-det-candidate-absolute-audit","seed":202610164100,"parameters":{"M":[[12,1],[2,1]],"candidate":10,"domain":"R_2x2_integer_slice","operation":"det-candidate-absolute-audit","semantic_template":"qf-density16-det-candidate-absolute-audit/{\"M\":[[12,1],[2,1]],\"candidate\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}