{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0001","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 0]]。求 det(M)。","answer":{"kind":"short_text","value":"0","explanation_zh":"det=0。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073001,"parameters":{"M":[[0,0],[0,0]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0002","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [2, 4]]。求 det(M)。","answer":{"kind":"short_text","value":"0","explanation_zh":"det=0。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073002,"parameters":{"M":[[1,2],[2,4]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"1\",\"2\"],[\"2\",\"4\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0003","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[0, 1], [1, 0]]。求 det(M)。","answer":{"kind":"short_text","value":"-1","explanation_zh":"det=-1。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073003,"parameters":{"M":[[0,1],[1,0]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"1\"],[\"1\",\"0\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0004","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-9, 4], [-2, 4]]。求 det(M)。","answer":{"kind":"short_text","value":"-28","explanation_zh":"det=-28。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073004,"parameters":{"M":[[-9,4],[-2,4]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0005","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-8, 1], [-2, 5]]。求 det(M)。","answer":{"kind":"short_text","value":"-38","explanation_zh":"det=-38。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073005,"parameters":{"M":[[-8,1],[-2,5]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0006","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-7, 2], [-2, 6]]。求 det(M)。","answer":{"kind":"short_text","value":"-38","explanation_zh":"det=-38。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073006,"parameters":{"M":[[-7,2],[-2,6]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0007","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-6, 3], [-2, 7]]。求 det(M)。","answer":{"kind":"short_text","value":"-36","explanation_zh":"det=-36。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073007,"parameters":{"M":[[-6,3],[-2,7]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-6\",\"3\"],[\"-2\",\"7\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0008","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-5, 4], [-2, 8]]。求 det(M)。","answer":{"kind":"short_text","value":"-32","explanation_zh":"det=-32。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073008,"parameters":{"M":[[-5,4],[-2,8]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-5\",\"4\"],[\"-2\",\"8\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0009","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-4, 1], [-2, 9]]。求 det(M)。","answer":{"kind":"short_text","value":"-34","explanation_zh":"det=-34。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073009,"parameters":{"M":[[-4,1],[-2,9]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0010","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-3, 2], [-2, 10]]。求 det(M)。","answer":{"kind":"short_text","value":"-26","explanation_zh":"det=-26。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073010,"parameters":{"M":[[-3,2],[-2,10]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0011","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-2, 3], [-2, 11]]。求 det(M)。","answer":{"kind":"short_text","value":"-16","explanation_zh":"det=-16。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073011,"parameters":{"M":[[-2,3],[-2,11]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0012","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[-1, 4], [-2, 12]]。求 det(M)。","answer":{"kind":"short_text","value":"-4","explanation_zh":"det=-4。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073012,"parameters":{"M":[[-1,4],[-2,12]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0013","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[0, 1], [-2, 13]]。求 det(M)。","answer":{"kind":"short_text","value":"2","explanation_zh":"det=2。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073013,"parameters":{"M":[[0,1],[-2,13]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"0\",\"1\"],[\"-2\",\"13\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0014","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [-2, 14]]。求 det(M)。","answer":{"kind":"short_text","value":"18","explanation_zh":"det=18。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073014,"parameters":{"M":[[1,2],[-2,14]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"1\",\"2\"],[\"-2\",\"14\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0015","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[2, 3], [-2, 15]]。求 det(M)。","answer":{"kind":"short_text","value":"36","explanation_zh":"det=36。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073015,"parameters":{"M":[[2,3],[-2,15]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"2\",\"3\"],[\"-2\",\"15\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0016","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[3, 4], [-2, 16]]。求 det(M)。","answer":{"kind":"short_text","value":"56","explanation_zh":"det=56。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073016,"parameters":{"M":[[3,4],[-2,16]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"3\",\"4\"],[\"-2\",\"16\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0017","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [-2, 17]]。求 det(M)。","answer":{"kind":"short_text","value":"70","explanation_zh":"det=70。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073017,"parameters":{"M":[[4,1],[-2,17]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"4\",\"1\"],[\"-2\",\"17\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0018","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[5, 2], [-2, 18]]。求 det(M)。","answer":{"kind":"short_text","value":"94","explanation_zh":"det=94。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073018,"parameters":{"M":[[5,2],[-2,18]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"5\",\"2\"],[\"-2\",\"18\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0019","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[6, 3], [-2, 19]]。求 det(M)。","answer":{"kind":"short_text","value":"120","explanation_zh":"det=120。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073019,"parameters":{"M":[[6,3],[-2,19]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"6\",\"3\"],[\"-2\",\"19\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0020","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[7, 4], [-2, 20]]。求 det(M)。","answer":{"kind":"short_text","value":"148","explanation_zh":"det=148。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073020,"parameters":{"M":[[7,4],[-2,20]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"7\",\"4\"],[\"-2\",\"20\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0021","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[8, 1], [-2, 21]]。求 det(M)。","answer":{"kind":"short_text","value":"170","explanation_zh":"det=170。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073021,"parameters":{"M":[[8,1],[-2,21]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"8\",\"1\"],[\"-2\",\"21\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0022","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[9, 2], [-2, 22]]。求 det(M)。","answer":{"kind":"short_text","value":"202","explanation_zh":"det=202。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073022,"parameters":{"M":[[9,2],[-2,22]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"9\",\"2\"],[\"-2\",\"22\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0023","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[10, 3], [-2, 23]]。求 det(M)。","answer":{"kind":"short_text","value":"236","explanation_zh":"det=236。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073023,"parameters":{"M":[[10,3],[-2,23]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"10\",\"3\"],[\"-2\",\"23\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0024","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[11, 4], [-2, 24]]。求 det(M)。","answer":{"kind":"short_text","value":"272","explanation_zh":"det=272。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073024,"parameters":{"M":[[11,4],[-2,24]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"11\",\"4\"],[\"-2\",\"24\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0025","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階行列式精確值","stem_zh":"實二階方陣，論域 R，M=[[12, 1], [-2, 25]]。求 det(M)。","answer":{"kind":"short_text","value":"302","explanation_zh":"det=302。二階值由主對角乘積減另一對角乘積，正、負、零均合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階行列式精確值"]},"variant":{"family_id":"qf-determinant-two-by-two","seed":202610073025,"parameters":{"M":[[12,1],[-2,25]],"operation":"det2","domain":"reals","semantic_template":"qf-determinant-two-by-two/{\"inputs\":[[\"12\",\"1\"],[\"-2\",\"25\"]],\"operation\":\"det2\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0026","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 0]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"否","explanation_zh":"det=0。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073026,"parameters":{"M":[[0,0],[0,0]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0027","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [2, 4]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"否","explanation_zh":"det=0。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073027,"parameters":{"M":[[1,2],[2,4]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"1\",\"2\"],[\"2\",\"4\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0028","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[0, 1], [1, 0]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-1。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073028,"parameters":{"M":[[0,1],[1,0]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"0\",\"1\"],[\"1\",\"0\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0029","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-9, 4], [-2, 4]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-28。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073029,"parameters":{"M":[[-9,4],[-2,4]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0030","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-8, 1], [-2, 5]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-38。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073030,"parameters":{"M":[[-8,1],[-2,5]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0031","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-7, 2], [-2, 6]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-38。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073031,"parameters":{"M":[[-7,2],[-2,6]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0034","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-4, 1], [-2, 9]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-34。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073034,"parameters":{"M":[[-4,1],[-2,9]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0035","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-3, 2], [-2, 10]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-26。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073035,"parameters":{"M":[[-3,2],[-2,10]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0036","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-2, 3], [-2, 11]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-16。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073036,"parameters":{"M":[[-2,3],[-2,11]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0037","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[-1, 4], [-2, 12]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=-4。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073037,"parameters":{"M":[[-1,4],[-2,12]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0039","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [-2, 14]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=18。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073039,"parameters":{"M":[[1,2],[-2,14]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"1\",\"2\"],[\"-2\",\"14\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0040","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[2, 3], [-2, 15]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=36。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073040,"parameters":{"M":[[2,3],[-2,15]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"2\",\"3\"],[\"-2\",\"15\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0041","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[3, 4], [-2, 16]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=56。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073041,"parameters":{"M":[[3,4],[-2,16]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"3\",\"4\"],[\"-2\",\"16\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0042","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [-2, 17]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=70。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073042,"parameters":{"M":[[4,1],[-2,17]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"4\",\"1\"],[\"-2\",\"17\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0043","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[5, 2], [-2, 18]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=94。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073043,"parameters":{"M":[[5,2],[-2,18]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"5\",\"2\"],[\"-2\",\"18\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0044","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[6, 3], [-2, 19]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=120。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073044,"parameters":{"M":[[6,3],[-2,19]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"6\",\"3\"],[\"-2\",\"19\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0045","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[7, 4], [-2, 20]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=148。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073045,"parameters":{"M":[[7,4],[-2,20]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"7\",\"4\"],[\"-2\",\"20\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0046","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[8, 1], [-2, 21]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=170。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073046,"parameters":{"M":[[8,1],[-2,21]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"8\",\"1\"],[\"-2\",\"21\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0047","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[9, 2], [-2, 22]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=202。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073047,"parameters":{"M":[[9,2],[-2,22]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"9\",\"2\"],[\"-2\",\"22\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0048","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[10, 3], [-2, 23]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=236。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073048,"parameters":{"M":[[10,3],[-2,23]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"10\",\"3\"],[\"-2\",\"23\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0049","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[11, 4], [-2, 24]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=272。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073049,"parameters":{"M":[[11,4],[-2,24]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"11\",\"4\"],[\"-2\",\"24\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0050","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"二階方陣的可逆資格","stem_zh":"實二階方陣，論域 R，M=[[12, 1], [-2, 25]]。M 在 R² 上是否可逆？","answer":{"kind":"short_text","value":"是","explanation_zh":"det=302。在此二階實矩陣模型，行列式非零才可逆；零行列式不能被忽略。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的可逆資格"]},"variant":{"family_id":"qf-determinant-invertibility","seed":202610073050,"parameters":{"M":[[12,1],[-2,25]],"operation":"invertible","domain":"reals","semantic_template":"qf-determinant-invertibility/{\"inputs\":[[\"12\",\"1\"],[\"-2\",\"25\"]],\"operation\":\"invertible\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0051","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[0, 0], [0, 0]]、B=[[1, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"0","explanation_zh":"乘積行列式=0。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073051,"parameters":{"A":[[0,0],[0,0]],"B":[[1,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"0\"]],[[\"1\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0052","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[1, 2], [2, 4]]、B=[[2, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"0","explanation_zh":"乘積行列式=0。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073052,"parameters":{"A":[[1,2],[2,4]],"B":[[2,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],[[\"2\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0053","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[0, 1], [1, 0]]、B=[[3, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-8","explanation_zh":"乘積行列式=-8。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073053,"parameters":{"A":[[0,1],[1,0]],"B":[[3,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[[\"3\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0054","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-9, 4], [-2, 4]]、B=[[4, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-224","explanation_zh":"乘積行列式=-224。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073054,"parameters":{"A":[[-9,4],[-2,4]],"B":[[4,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[[\"4\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0055","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-8, 1], [-2, 5]]、B=[[5, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-418","explanation_zh":"乘積行列式=-418。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073055,"parameters":{"A":[[-8,1],[-2,5]],"B":[[5,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[[\"5\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0056","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-7, 2], [-2, 6]]、B=[[6, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-532","explanation_zh":"乘積行列式=-532。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073056,"parameters":{"A":[[-7,2],[-2,6]],"B":[[6,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[[\"6\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0057","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-6, 3], [-2, 7]]、B=[[7, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-504","explanation_zh":"乘積行列式=-504。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073057,"parameters":{"A":[[-6,3],[-2,7]],"B":[[7,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[[\"7\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0058","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-5, 4], [-2, 8]]、B=[[8, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-544","explanation_zh":"乘積行列式=-544。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073058,"parameters":{"A":[[-5,4],[-2,8]],"B":[[8,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[[\"8\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0059","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-4, 1], [-2, 9]]、B=[[9, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-680","explanation_zh":"乘積行列式=-680。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073059,"parameters":{"A":[[-4,1],[-2,9]],"B":[[9,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[[\"9\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0060","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-3, 2], [-2, 10]]、B=[[10, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-520","explanation_zh":"乘積行列式=-520。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073060,"parameters":{"A":[[-3,2],[-2,10]],"B":[[10,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[[\"10\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0061","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-2, 3], [-2, 11]]、B=[[11, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-368","explanation_zh":"乘積行列式=-368。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073061,"parameters":{"A":[[-2,3],[-2,11]],"B":[[11,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[[\"11\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0062","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[-1, 4], [-2, 12]]、B=[[12, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"-104","explanation_zh":"乘積行列式=-104。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073062,"parameters":{"A":[[-1,4],[-2,12]],"B":[[12,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[[\"12\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0063","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[0, 1], [-2, 13]]、B=[[13, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"52","explanation_zh":"乘積行列式=52。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073063,"parameters":{"A":[[0,1],[-2,13]],"B":[[13,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[[\"13\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0064","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[1, 2], [-2, 14]]、B=[[14, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"522","explanation_zh":"乘積行列式=522。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073064,"parameters":{"A":[[1,2],[-2,14]],"B":[[14,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[[\"14\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0065","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[2, 3], [-2, 15]]、B=[[15, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"1152","explanation_zh":"乘積行列式=1152。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073065,"parameters":{"A":[[2,3],[-2,15]],"B":[[15,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"15\",\"-1\"],[\"2\",\"2\"]],[[\"2\",\"3\"],[\"-2\",\"15\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0066","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[3, 4], [-2, 16]]、B=[[16, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"1792","explanation_zh":"乘積行列式=1792。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073066,"parameters":{"A":[[3,4],[-2,16]],"B":[[16,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"16\",\"-1\"],[\"0\",\"2\"]],[[\"3\",\"4\"],[\"-2\",\"16\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0067","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[4, 1], [-2, 17]]、B=[[17, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"2450","explanation_zh":"乘積行列式=2450。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073067,"parameters":{"A":[[4,1],[-2,17]],"B":[[17,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"17\",\"-1\"],[\"1\",\"2\"]],[[\"4\",\"1\"],[\"-2\",\"17\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0068","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[5, 2], [-2, 18]]、B=[[18, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"3572","explanation_zh":"乘積行列式=3572。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073068,"parameters":{"A":[[5,2],[-2,18]],"B":[[18,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"18\",\"-1\"],[\"2\",\"2\"]],[[\"5\",\"2\"],[\"-2\",\"18\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0069","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[6, 3], [-2, 19]]、B=[[19, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"4560","explanation_zh":"乘積行列式=4560。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073069,"parameters":{"A":[[6,3],[-2,19]],"B":[[19,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"19\",\"-1\"],[\"0\",\"2\"]],[[\"6\",\"3\"],[\"-2\",\"19\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0070","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[7, 4], [-2, 20]]、B=[[20, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"6068","explanation_zh":"乘積行列式=6068。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073070,"parameters":{"A":[[7,4],[-2,20]],"B":[[20,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"20\",\"-1\"],[\"1\",\"2\"]],[[\"7\",\"4\"],[\"-2\",\"20\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0071","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[8, 1], [-2, 21]]、B=[[21, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"7480","explanation_zh":"乘積行列式=7480。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073071,"parameters":{"A":[[8,1],[-2,21]],"B":[[21,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"21\",\"-1\"],[\"2\",\"2\"]],[[\"8\",\"1\"],[\"-2\",\"21\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0072","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[9, 2], [-2, 22]]、B=[[22, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"8888","explanation_zh":"乘積行列式=8888。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073072,"parameters":{"A":[[9,2],[-2,22]],"B":[[22,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"22\",\"-1\"],[\"0\",\"2\"]],[[\"9\",\"2\"],[\"-2\",\"22\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0073","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[10, 3], [-2, 23]]、B=[[23, -1], [1, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"11092","explanation_zh":"乘積行列式=11092。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073073,"parameters":{"A":[[10,3],[-2,23]],"B":[[23,-1],[1,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[[\"23\",\"-1\"],[\"1\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0074","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[11, 4], [-2, 24]]、B=[[24, -1], [2, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"13600","explanation_zh":"乘積行列式=13600。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073074,"parameters":{"A":[[11,4],[-2,24]],"B":[[24,-1],[2,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[[\"24\",\"-1\"],[\"2\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0075","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"矩陣乘積的行列式","stem_zh":"實二階方陣，論域 R，A=[[12, 1], [-2, 25]]、B=[[25, -1], [0, 2]]。求 det(AB)，不要把矩陣乘法誤當逐元素乘法。","answer":{"kind":"short_text","value":"15100","explanation_zh":"乘積行列式=15100。本產生端用 det(A)det(B)，主端另行核對乘積矩陣；不宣稱兩矩陣本身可交換。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣乘積的行列式"]},"variant":{"family_id":"qf-determinant-product","seed":202610073075,"parameters":{"A":[[12,1],[-2,25]],"B":[[25,-1],[0,2]],"operation":"product_determinant","domain":"reals","semantic_template":"qf-determinant-product/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[[\"25\",\"-1\"],[\"0\",\"2\"]]],\"operation\":\"product_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0076","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 0]]。給 c=0，求 det(cM)。","answer":{"kind":"short_text","value":"0","explanation_zh":"縮放行列式=0。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073076,"parameters":{"M":[[0,0],[0,0]],"c":0,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[\"zero_scalar\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0077","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [2, 4]]。給 c=-1，求 det(cM)。","answer":{"kind":"short_text","value":"0","explanation_zh":"縮放行列式=0。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073077,"parameters":{"M":[[1,2],[2,4]],"c":-1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0078","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[0, 1], [1, 0]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"-9","explanation_zh":"縮放行列式=-9。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073078,"parameters":{"M":[[0,1],[1,0]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0079","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-9, 4], [-2, 4]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"-448","explanation_zh":"縮放行列式=-448。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073079,"parameters":{"M":[[-9,4],[-2,4]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0080","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-8, 1], [-2, 5]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"-38","explanation_zh":"縮放行列式=-38。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073080,"parameters":{"M":[[-8,1],[-2,5]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0081","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-7, 2], [-2, 6]]。給 c=-2，求 det(cM)。","answer":{"kind":"short_text","value":"-152","explanation_zh":"縮放行列式=-152。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073081,"parameters":{"M":[[-7,2],[-2,6]],"c":-2,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],\"2\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0082","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-6, 3], [-2, 7]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"-324","explanation_zh":"縮放行列式=-324。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073082,"parameters":{"M":[[-6,3],[-2,7]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0083","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-5, 4], [-2, 8]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"-512","explanation_zh":"縮放行列式=-512。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073083,"parameters":{"M":[[-5,4],[-2,8]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0084","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-4, 1], [-2, 9]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"-34","explanation_zh":"縮放行列式=-34。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073084,"parameters":{"M":[[-4,1],[-2,9]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0085","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-3, 2], [-2, 10]]。給 c=-2，求 det(cM)。","answer":{"kind":"short_text","value":"-104","explanation_zh":"縮放行列式=-104。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073085,"parameters":{"M":[[-3,2],[-2,10]],"c":-2,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],\"2\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0086","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-2, 3], [-2, 11]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"-144","explanation_zh":"縮放行列式=-144。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073086,"parameters":{"M":[[-2,3],[-2,11]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0087","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[-1, 4], [-2, 12]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"-64","explanation_zh":"縮放行列式=-64。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073087,"parameters":{"M":[[-1,4],[-2,12]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0088","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[0, 1], [-2, 13]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"2","explanation_zh":"縮放行列式=2。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073088,"parameters":{"M":[[0,1],[-2,13]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0089","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[1, 2], [-2, 14]]。給 c=-2，求 det(cM)。","answer":{"kind":"short_text","value":"72","explanation_zh":"縮放行列式=72。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073089,"parameters":{"M":[[1,2],[-2,14]],"c":-2,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],\"2\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0090","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[2, 3], [-2, 15]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"324","explanation_zh":"縮放行列式=324。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073090,"parameters":{"M":[[2,3],[-2,15]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0091","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[3, 4], [-2, 16]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"896","explanation_zh":"縮放行列式=896。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073091,"parameters":{"M":[[3,4],[-2,16]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0092","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [-2, 17]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"70","explanation_zh":"縮放行列式=70。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073092,"parameters":{"M":[[4,1],[-2,17]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0093","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[5, 2], [-2, 18]]。給 c=-2，求 det(cM)。","answer":{"kind":"short_text","value":"376","explanation_zh":"縮放行列式=376。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073093,"parameters":{"M":[[5,2],[-2,18]],"c":-2,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],\"2\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0094","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[6, 3], [-2, 19]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"1080","explanation_zh":"縮放行列式=1080。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073094,"parameters":{"M":[[6,3],[-2,19]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0095","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[7, 4], [-2, 20]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"2368","explanation_zh":"縮放行列式=2368。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073095,"parameters":{"M":[[7,4],[-2,20]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0096","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[8, 1], [-2, 21]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"170","explanation_zh":"縮放行列式=170。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073096,"parameters":{"M":[[8,1],[-2,21]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0097","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[9, 2], [-2, 22]]。給 c=-2，求 det(cM)。","answer":{"kind":"short_text","value":"808","explanation_zh":"縮放行列式=808。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073097,"parameters":{"M":[[9,2],[-2,22]],"c":-2,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],\"2\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0098","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[10, 3], [-2, 23]]。給 c=3，求 det(cM)。","answer":{"kind":"short_text","value":"2124","explanation_zh":"縮放行列式=2124。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073098,"parameters":{"M":[[10,3],[-2,23]],"c":3,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],\"3\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0099","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[11, 4], [-2, 24]]。給 c=-4，求 det(cM)。","answer":{"kind":"short_text","value":"4352","explanation_zh":"縮放行列式=4352。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073099,"parameters":{"M":[[11,4],[-2,24]],"c":-4,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],\"4\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r03-determinant-0100","batch_id":"batch-20261006-advance-20261007-r03-determinant-foundations","language":"zh-Hant","source_mko_ids":["mko-determinant"],"title_zh":"純量縮放的二階行列式","stem_zh":"實二階方陣，論域 R，M=[[12, 1], [-2, 25]]。給 c=1，求 det(cM)。","answer":{"kind":"short_text","value":"302","explanation_zh":"縮放行列式=302。本產生端使用 c²det(M)，不是只乘 c；二維負縮放也帶平方因子。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["純量縮放的二階行列式"]},"variant":{"family_id":"qf-determinant-scalar","seed":202610073100,"parameters":{"M":[[12,1],[-2,25]],"c":1,"operation":"scaled_determinant","domain":"reals","semantic_template":"qf-determinant-scalar/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],\"1\"],\"operation\":\"scaled_determinant\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}