{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r03-determinant-0029",
  "pilot_number": "R0061-31229",
  "number": 31229,
  "family_id": "qf-determinant-invertibility",
  "title": {
    "zh-Hant": "第 31229 題：二階方陣的可逆資格",
    "en": "Question 31229: Invertibility over the Real Field"
  },
  "original_question": {
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261006-advance-20261007-r03-determinant-0029",
    "batch_id": "batch-20261006-advance-20261007-r03-determinant-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "beabaae0f65b3b15c5733e7cad7f93d2896b6dcb2eb663499dd9aaba77760e86",
      "family_id": "qf-determinant-invertibility",
      "seed": 202610073029,
      "semantic_template": "qf-determinant-invertibility/{\"inputs\":[[\"-9\",\"4\"],[\"-2\",\"4\"]],\"operation\":\"invertible\"}"
    },
    "fields": {
      "title": "Invertibility over the Real Field",
      "stem": "Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The determinant is -28. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring."
      },
      "skills": [
        "Invertibility over the Real Field"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "實二階方陣，論域 R，M=[[-9, 4], [-2, 4]]。M 在 R² 上是否可逆？",
        "en": "Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-determinant-invertibility, invertible, domain, reals]",
      "after": "derived = [[real_invertibility_YesNo]]",
      "relation": "implies",
      "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。",
        "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Det requested matrix ORIGINAL main diagonal product := mul(-9, 4)",
      "after": "Det requested matrix ORIGINAL main diagonal product = -36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Det requested matrix ORIGINAL main diagonal product，不以小數近似替代。",
          "en": "Recompute Det requested matrix ORIGINAL main diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-9",
          "4"
        ],
        "output": "-36",
        "label": "Det requested matrix ORIGINAL main diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Det requested matrix ORIGINAL other diagonal product := mul(4, -2)",
      "after": "Det requested matrix ORIGINAL other diagonal product = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Det requested matrix ORIGINAL other diagonal product，不以小數近似替代。",
          "en": "Recompute Det requested matrix ORIGINAL other diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "-2"
        ],
        "output": "-8",
        "label": "Det requested matrix ORIGINAL other diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Det requested matrix ORIGINAL signed ad minus bc := sub(-36, -8)",
      "after": "Det requested matrix ORIGINAL signed ad minus bc = -28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序，作精確整數計算；只對原始輸入檢查SAFE，衍生整數不再套上輸入界，且不用浮點近似。",
        "en": "Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Det requested matrix ORIGINAL signed ad minus bc，不以小數近似替代。",
          "en": "Recompute Det requested matrix ORIGINAL signed ad minus bc with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-36",
          "-8"
        ],
        "output": "-28",
        "label": "Det requested matrix ORIGINAL signed ad minus bc"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [Det requested matrix, [[-9, 4], [-2, 4]]]",
      "after": "derived = [-36, -8, -28]",
      "relation": "implies",
      "rule_id": "Determinant_2x2_signed_ad_MINUS_bc",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對此2×2矩陣明列ad、bc，再用ad−bc保符號；不是永久式ad+bc。衍生矩陣元素不重套SAFE。",
        "en": "For THIS2×2 matrix explicitly calculate ad and bc,then retain the sign in ad−bc,not the permanent ad+bc.Derived entries are not SAFE-recapped."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [[[-9, 4], [-2, 4]], -28]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Determinant_REAL_field_invertible_IFF_NONZERO_not_integer_unit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在R域2×2可逆當且僅當det非0，非只准±1。可用伴隨矩陣除det回代檢查；det0不能除。",
        "en": "Over the REAL field,a2×2 matrix is invertible IFF its determinant is NONZERO,not only±1.The adjugate divided by det supplies a reverse check;never divide by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。",
          "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "是"
  },
  "conclusion": {
    "canonical_value": "是",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261006-advance-20261007-r03-determinant-foundations",
      "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r03-determinant-foundations/questions.jsonl",
      "batch_sha256": "f15de01e2746af1ac9890f8b9ffb67551ce947d201efb14cc15f52534f931f0d",
      "original_line": 29,
      "original_line_sha256": "5c6f9e081954b0103242dfc008503f7c3334463ad9d3344ab56ccc27d899af7e",
      "question_compact_sha256": "beabaae0f65b3b15c5733e7cad7f93d2896b6dcb2eb663499dd9aaba77760e86",
      "english_row_compact_sha256": "f7f4280025673ae2ca4152cb57cc004deb44fa5074411c2d81514589684312fc",
      "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r03-determinant-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r03-determinant-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-determinant": {
        "sha256": "d49050d70b58e0d9d2cab2f5ddc64133813f5a99418865b5941af4f1ec240093",
        "english_sha256": "c20e33f6fef226f483add30aa030f5fa68b5878d5ea16a6e3e75b0b360299311",
        "original_path": "public/data/mko/determinant.json",
        "english_path": "public/data/i18n/editions/mko-determinant.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c",
    "original_candidate_sha256": "962237d27f595792e98f71f42f869d01f5e23edf563af92adcc5e94794c040a0",
    "original_candidate_line_sha256": "962237d27f595792e98f71f42f869d01f5e23edf563af92adcc5e94794c040a0"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-011-p62",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69",
      "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050",
      "arithmetic_steps": 3
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}