{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-determinant",
  "semantic_family_id": "sf-determinant",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實二階方陣的行列式"
  },
  "summary": {
    "zh-Hant": "2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。"
  },
  "statement": {
    "zh-Hant": "對實矩陣 M=[[a,b],[c,d]]，本輪 D(M)=det(M)=ad−bc。M 在實數域可逆當且僅當 det(M)≠0；對同為2×2的實矩陣 A、B，det(AB)=det(A)det(B)，且 det(tM)=t²det(M)。"
  },
  "formula": {
    "tex": "D(M)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "M",
          "glyph": "M"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "abff79594cb3f86a316679bf964f035574e7d0aa706447e51ecf6306bc71cea8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "矩陣是2×2方陣，數域是實數；不把實數域非零即可逆的敘述原樣搬到整數環。"
    },
    {
      "id": "a2",
      "zh-Hant": "整數元素及整數 t 是精確題庫切片；負的 t 也由 t² 作用，t=0 使縮放矩陣行列式零。"
    },
    {
      "id": "a3",
      "zh-Hant": "AB 是通常的行乘列矩陣乘積，不是逐項相乘；乘積公式不表示 A、B 可交換。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "two_by_two_determinant",
      "role_zh": "本輪二階行列式",
      "scope": "representative_formula"
    },
    {
      "token": "M",
      "role": "real_square_matrix",
      "role_zh": "2×2實方陣",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-matrix",
      "title_zh": "有限矩陣的尺寸、索引與轉置",
      "reason_zh": "先驗方陣尺寸及項的位置。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "可逆判準明示實數域。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "符號與位置都重要，不能把 ad−bc 改成 ad+bc 或對負行列式取絕對值。非零行列式包含正與負兩種，不必等於一。",
      "實數域可逆性是此切片的精確條件；矩陣元素恰為整數不代表它在整數環也有整數反矩陣。",
      "主端以實際 AB 與 tM 重新算行列式；出題端可用乘積與 t² 恆等式形成另一算法。這些是有限計算驗收，不新增一般高階行列式 Evidence。"
    ],
    "common_misconception_zh": "非零行列式必須為正、把逐項乘法叫 AB，或忽略可逆判準的數域。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "二階整數行列式切片可重算；一般階數、行列式理論及域／環區別的形式橋待完成。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "固定版 det_fin_two、det_mul、det_smul 及實數域 nonsingular inverse 橋接。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-determinant.md 整理；2026-10-06 脈衝 R03（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}