{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r07-eigenvalue-0008",
  "pilot_number": "R0006-03708",
  "number": 3708,
  "family_id": "qf-density-eigen-characteristic-coefficients",
  "title": {
    "zh-Hant": "第 3708 題：實二階矩陣的降冪特徵多項式",
    "en": "Question 3708: Descending Characteristic Coefficients of a Real 2×2 Matrix"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r07-eigenvalue-0008",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-eigenvalue"
    ],
    "title_zh": "實二階矩陣的降冪特徵多項式",
    "stem_zh": "實二階方陣與精確整數資料。M=[[-5,2],[-1,8]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。",
    "answer": {
      "kind": "expression",
      "value": "(1, -3, -38)",
      "explanation_zh": "降冪特徵多項式係數=(1, -3, -38)。計算1、−trace、det的指定二階式，不推論對角化或一般譜定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "實二階矩陣的降冪特徵多項式"
      ]
    },
    "variant": {
      "family_id": "qf-density-eigen-characteristic-coefficients",
      "seed": 202610097008,
      "parameters": {
        "M": [
          [
            -5,
            2
          ],
          [
            -1,
            8
          ]
        ],
        "operation": "density_eigen_characteristic_coefficients",
        "domain": "real_two_by_two",
        "semantic_template": "qf-density-eigen-characteristic-coefficients/[\"3\",\"-38\"]"
      }
    },
    "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": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r07-eigenvalue-0008",
    "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1b47162ab85c0fbfefaf35c79921a7a8e71a28a235a8d110d17b5258ecd36936",
      "family_id": "qf-density-eigen-characteristic-coefficients",
      "seed": 202610097008,
      "semantic_template": "qf-density-eigen-characteristic-coefficients/[\"3\",\"-38\"]"
    },
    "fields": {
      "title": "Descending Characteristic Coefficients of a Real 2×2 Matrix",
      "stem": "Over R, M=[[-5, 2], [-1, 8]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).",
      "answer": {
        "kind": "expression",
        "value": "(1, -3, -38)",
        "explanation": "Use (1,−tr(M),det(M)) to obtain (1, -3, -38), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."
      },
      "skills": [
        "Descending Characteristic Coefficients of a Real 2×2 Matrix"
      ],
      "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": "實二階方陣與精確整數資料。M=[[-5,2],[-1,8]]。列 det(tI−M) 的降冪係數 (t²係數, t係數, 常數)，採整數有序三元組。",
        "en": "Over R, M=[[-5, 2], [-1, 8]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[-5, 2], [-1, 8]]",
      "after": "exact real2×2 matrix;no spectrum or diagonalization inferred",
      "relation": "implies",
      "rule_id": "two_by_two_matrix_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只處理宣告的實二階矩陣；係數、判別式或單一候選殘差都不自動給出完整譜或可對角化性。",
        "en": "Use the declared real2×2 matrix;coefficients,a discriminant or one candidate residual does not automatically establish the entire spectrum or diagonalizability."
      },
      "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": "trace := sum(-5, 8)",
      "after": "trace = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "跡為主對角線之和。",
        "en": "The trace is the sum of diagonal entries."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 trace，不以小數近似替代。",
          "en": "Recompute trace with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "8"
        ],
        "output": "3",
        "label": "trace"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "diagonal product := mul(-5, 8)",
      "after": "diagonal product = -40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式第一項a·d。",
        "en": "Compute the determinant term a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 diagonal product，不以小數近似替代。",
          "en": "Recompute diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "8"
        ],
        "output": "-40",
        "label": "diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "off-diagonal product := mul(2, -1)",
      "after": "off-diagonal product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二項為b·c，後續要減而非加。",
        "en": "Compute b·c,which must be subtracted rather than added."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 off-diagonal product，不以小數近似替代。",
          "en": "Recompute off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "off-diagonal product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "determinant := sub(-40, -2)",
      "after": "determinant = -38",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確計算ad−bc。",
        "en": "Compute ad−bc exactly."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 determinant，不以小數近似替代。",
          "en": "Recompute determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-40",
          "-2"
        ],
        "output": "-38",
        "label": "determinant"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "negative trace := mul(-1, 3)",
      "after": "negative trace = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "det(tI−M)的線性項係數是−trace，保留負號。",
        "en": "The linear coefficient of det(tI−M) is−trace,including its sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 negative trace，不以小數近似替代。",
          "en": "Recompute negative trace with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "3"
        ],
        "output": "-3",
        "label": "negative trace"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "det(tI−M)=(t−-5)(t−8)−(2)(-1)",
      "after": "t²+(-3)t+(-38)",
      "relation": "implies",
      "rule_id": "characteristic_polynomial_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "展開二階行列式得到降冪係數(1,−trace,det)；維持欄序及線性項的負跡，不由係數逕推可對角化性。",
        "en": "Expand the2×2 determinant to get descending coefficients(1,−trace,det).Keep this order and the negative trace;the coefficients alone do not establish diagonalizability."
      },
      "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": [
      1,
      "-3",
      "-38"
    ],
    "trace": "3",
    "determinant": "-38"
  },
  "conclusion": {
    "canonical_value": "(1, -3, -38)",
    "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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    "question_binding": {
      "batch_id": "batch-20261007-advance-20261009-r07-eigenvalue-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r07-eigenvalue-foundations/questions.jsonl",
      "batch_sha256": "e7cddab340faf1ec3ec8afd1de71da0f2940357248570ee9b85309e2cdfda2be",
      "original_line": 8,
      "original_line_sha256": "bd2bb1517177433e4a228a286227395f89427cb762e772264b5b325e7de00220",
      "question_compact_sha256": "1b47162ab85c0fbfefaf35c79921a7a8e71a28a235a8d110d17b5258ecd36936",
      "english_row_compact_sha256": "7fa82507a25ad6c8a632714399d38ae9b0d595481c27c118f41dc4a23dd9a48d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r07-eigenvalue-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r07-eigenvalue-foundations.en.jsonl"
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    "mko_bindings": {
      "mko-eigenvalue": {
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        "original_path": "public/data/mko/eigenvalue.json",
        "english_path": "public/data/i18n/editions/mko-eigenvalue.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "d031bb5584c2a227a27b53e98037c307b27481ddf51aa0ee2388d3752eec019a",
    "original_candidate_line_sha256": "d031bb5584c2a227a27b53e98037c307b27481ddf51aa0ee2388d3752eec019a"
  },
  "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-20261026-004-p7",
    "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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      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f00c94f7e8e85a074a99502efcf566ef1f3975de0df02e4de8dfa0fde50cac81",
      "arithmetic_steps": 5
    },
    "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."
  }
}