{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r04-determinant-0013",
  "pilot_number": "R0013-07363",
  "number": 7363,
  "family_id": "qf-density16-det-row-swap-signed",
  "title": {
    "zh-Hant": "第 7363 題：交換兩行的有號行列式",
    "en": "Question 7363: Signed Determinants Before and After a Row Swap"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r04-determinant-0013",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-determinant"
    ],
    "title_zh": "交換兩行的有號行列式",
    "stem_zh": "R上2×2矩陣，D為有號行列式。M=[[0, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)",
      "explanation_zh": "有限結構核對=([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)。保留行列位置及正負號，交換兩行再核二階乘積差。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "交換兩行的有號行列式"
      ]
    },
    "variant": {
      "family_id": "qf-density16-det-row-swap-signed",
      "seed": 202610164013,
      "parameters": {
        "M": [
          [
            0,
            1
          ],
          [
            2,
            1
          ]
        ],
        "domain": "R_2x2_integer_slice",
        "operation": "det-row-swap-signed",
        "semantic_template": "qf-density16-det-row-swap-signed/{\"M\":[[0,1],[2,1]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r04-determinant-0013",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1a10ac3a7d7d743622845865722c1890075d716225f8065d4e7cc2568cd3c153",
      "family_id": "qf-density16-det-row-swap-signed",
      "seed": 202610164013,
      "semantic_template": "qf-density16-det-row-swap-signed/{\"M\":[[0,1],[2,1]]}"
    },
    "fields": {
      "title": "Signed Determinants Before and After a Row Swap",
      "stem": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)",
        "explanation": "Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)."
      },
      "skills": [
        "Signed Determinants Before and After a Row Swap"
      ],
      "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": "R上2×2矩陣，D為有號行列式。M=[[0, 1], [2, 1]]。求 (M, D(M), 交換兩行的矩陣, 交換後D, 兩個D之和)。",
        "en": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[0, 1], [2, 1]]",
      "after": "real2×2 integer-entry slice;determinants are signed",
      "relation": "implies",
      "rule_id": "signed_two_by_two_determinant_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式按有號ad−bc計算，不以絕對值取代；矩陣乘法按行乘列，不當逐項乘法。此有限核對不是一般形式證明。",
        "en": "Compute the signed determinant ad−bc,not its absolute value;matrix multiplication uses row-by-column,not entrywise products.This finite audit is not a general formal proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "M diagonal product := mul(0, 1)",
      "after": "M diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M diagonal product，不以小數近似替代。",
          "en": "Recompute M diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "M diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "M off-diagonal product := mul(1, 2)",
      "after": "M off-diagonal product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M off-diagonal product，不以小數近似替代。",
          "en": "Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "M off-diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "M signed determinant := sub(0, 2)",
      "after": "M signed determinant = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M signed determinant，不以小數近似替代。",
          "en": "Recompute M signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "2"
        ],
        "output": "-2",
        "label": "M signed determinant"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "row-swapped M diagonal product := mul(2, 1)",
      "after": "row-swapped M diagonal product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row-swapped M diagonal product，不以小數近似替代。",
          "en": "Recompute row-swapped M diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "row-swapped M diagonal product"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "row-swapped M off-diagonal product := mul(1, 0)",
      "after": "row-swapped M off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row-swapped M off-diagonal product，不以小數近似替代。",
          "en": "Recompute row-swapped M off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "row-swapped M off-diagonal product"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "row-swapped M signed determinant := sub(2, 0)",
      "after": "row-swapped M signed determinant = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 row-swapped M signed determinant，不以小數近似替代。",
          "en": "Recompute row-swapped M signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "row-swapped M signed determinant"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original plus row-swapped determinants := sum(-2, 2)",
      "after": "original plus row-swapped determinants = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交換兩行反號，兩個行列式和應為0。",
        "en": "A row swap reverses the sign,so the two determinants sum to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original plus row-swapped determinants，不以小數近似替代。",
          "en": "Recompute original plus row-swapped determinants with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "2"
        ],
        "output": "0",
        "label": "original plus row-swapped determinants"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "M=[[0, 1], [2, 1]];swapped rows=[[2, 1], [0, 1]]",
      "after": "signed determinants=(-2,2)",
      "relation": "implies",
      "rule_id": "determinant_row_swap_sign",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交換的是行而非轉置；det=0時反號仍0，不能因此把符號規則改成絕對值。",
        "en": "This swaps rows rather than transposing;when det=0 its negative is still0,which does not replace the sign rule by absolute values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          0,
          1
        ],
        [
          2,
          1
        ]
      ],
      "-2",
      [
        [
          2,
          1
        ],
        [
          0,
          1
        ]
      ],
      "2",
      "0"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)",
    "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": {
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    },
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    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "fadc03b711e43dd4e1a1f5f168f6d0c3ede9f3e1ee34161b4e606bb0daad92b4",
    "original_candidate_line_sha256": "fadc03b711e43dd4e1a1f5f168f6d0c3ede9f3e1ee34161b4e606bb0daad92b4"
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "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": {
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    "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": {
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}