{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r01-matrix-rank-0090",
  "pilot_number": "R0034-17615",
  "number": 17615,
  "family_id": "qf-mod34-rank-determinant-zero-boundary",
  "title": {
    "zh-Hant": "第 17615 題：矩陣秩",
    "en": "Question 17615: Zero Determinant Does Not Determine Which Lower Rank Occurs"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r01-matrix-rank-0090",
    "batch_id": "batch-20261014-advance-20261103-r01-matrix-rank-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-matrix-rank"
    ],
    "title_zh": "矩陣秩",
    "stem_zh": "2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,15],[2,30]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)",
      "explanation_zh": "Wave5第三層核對=(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "矩陣秩"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-rank-determinant-zero-boundary",
      "seed": 202611031090,
      "parameters": {
        "matrix": [
          [
            1,
            15
          ],
          [
            2,
            30
          ]
        ],
        "domain": "Exact_square_matrix_rank",
        "operation": "rank-determinant-zero-boundary",
        "semantic_template": "qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,15],[2,30]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r01-matrix-rank-0090",
    "batch_id": "batch-20261014-advance-20261103-r01-matrix-rank-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7f6d4904509fc5186a39040cd9ed2ce2e3cdce3ebcdffbfdaccbbb252f6cd8fe",
      "family_id": "qf-mod34-rank-determinant-zero-boundary",
      "seed": 202611031090,
      "semantic_template": "qf-mod34-rank-determinant-zero-boundary/{\"matrix\":[[1,15],[2,30]]}"
    },
    "fields": {
      "title": "Zero Determinant Does Not Determine Which Lower Rank Occurs",
      "stem": "Use the complete ordered integer matrix A=[[1, 15], [2, 30]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.",
      "answer": {
        "kind": "expression",
        "value": "(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)",
        "explanation": "Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)."
      },
      "skills": [
        "Zero Determinant Does Not Determine Which Lower Rank Occurs"
      ],
      "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": "2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-determinant-zero-boundary；資料={\"matrix\":[[1,15],[2,30]]}。請依 [determinant,zero,rank,consequence,boundary] 欄序回答。",
        "en": "Use the complete ordered integer matrix A=[[1, 15], [2, 30]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "rank pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "rank pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute rank pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "rank pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "rank pivot0 column0 normalized entry1 := div(15, 1)",
      "after": "rank pivot0 column0 normalized entry1 = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot0 column0 normalized entry1，不以小數近似替代。",
          "en": "Recompute rank pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "1"
        ],
        "output": "15",
        "label": "rank pivot0 column0 normalized entry1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 15]",
      "after": "normalized full row=[1, 15]",
      "relation": "implies",
      "rule_id": "rank_nonzero_pivot_entire_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，逐格保留精確有理值；乘回可逆，禁止浮點近似。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rationals.The inverse operation multiplies back;do not approximate by floats."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "rank eliminate row1 pivot0 column0 entry0 product := mul(2, 1)",
      "after": "rank eliminate row1 pivot0 column0 entry0 product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row1 pivot0 column0 entry0 product，不以小數近似替代。",
          "en": "Recompute rank eliminate row1 pivot0 column0 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "rank eliminate row1 pivot0 column0 entry0 product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "rank eliminate row1 pivot0 column0 entry0 difference := sub(2, 2)",
      "after": "rank eliminate row1 pivot0 column0 entry0 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row1 pivot0 column0 entry0 difference，不以小數近似替代。",
          "en": "Recompute rank eliminate row1 pivot0 column0 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "rank eliminate row1 pivot0 column0 entry0 difference"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "rank eliminate row1 pivot0 column0 entry1 product := mul(2, 15)",
      "after": "rank eliminate row1 pivot0 column0 entry1 product = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row1 pivot0 column0 entry1 product，不以小數近似替代。",
          "en": "Recompute rank eliminate row1 pivot0 column0 entry1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "15"
        ],
        "output": "30",
        "label": "rank eliminate row1 pivot0 column0 entry1 product"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "rank eliminate row1 pivot0 column0 entry1 difference := sub(30, 30)",
      "after": "rank eliminate row1 pivot0 column0 entry1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row1 pivot0 column0 entry1 difference，不以小數近似替代。",
          "en": "Recompute rank eliminate row1 pivot0 column0 entry1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "30",
          "30"
        ],
        "output": "0",
        "label": "rank eliminate row1 pivot0 column0 entry1 difference"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "target row=1;old row=[2, 30];subtract factor=2 times pivot row=[1, 15]",
      "after": "new row=[0, 0];cleared pivot-column entry=0",
      "relation": "implies",
      "rule_id": "rank_exact_all_entry_elimination_not_partial_trace",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每格作原值減factor倍pivot格，加回即可反轉；把此pivot欄其他列都消零才繼續。",
        "en": "Subtract the factor times the pivot entry in every coordinate;adding back reverses the operation.Clear this pivot column in all other rows before continuing."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "whole matrix column=1;unused row indices=[1];remaining column values=[0]",
      "after": "no nonzero pivot;current pivot count=1",
      "relation": "implies",
      "rule_id": "rank_zero_remaining_column_no_pivot",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只在尚未使用列找pivot，皆零則此欄不增秩。",
        "en": "Search unused rows only;an all-zero remainder adds no pivot."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "complete original square matrix=[[1, 15], [2, 30]];final exact RREF=[[1, 15], [0, 0]];increasing pivot columns=[0]",
      "after": "all rows retained inclzero;rank=1;row operations invertible;row-rank equals column-rank by the finite-dimensional theorem",
      "relation": "implies",
      "rule_id": "rank_full_rref_rank_preserving_certificate_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整消去保留所有列，非零pivot數給精確rank。列運算保rank；row-rank=column-rank依有限維定理，不以幾個成功樣本證一般定理，亦不宣稱實際欄空間逐點不變。",
        "en": "Keep every row and complete all available pivots for exact rank.Invertible row operations preserve rank.Row-rank equals column-rank by the finite-dimensional theorem,not a few successful samples;literal column spans need not remain unchanged."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "whole original matrix diagonal product := mul(1, 30)",
      "after": "whole original matrix diagonal product = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix diagonal product，不以小數近似替代。",
          "en": "Recompute whole original matrix diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "30"
        ],
        "output": "30",
        "label": "whole original matrix diagonal product"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "whole original matrix off-diagonal product := mul(15, 2)",
      "after": "whole original matrix off-diagonal product = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix off-diagonal product，不以小數近似替代。",
          "en": "Recompute whole original matrix off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "2"
        ],
        "output": "30",
        "label": "whole original matrix off-diagonal product"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "whole original matrix determinant difference := sub(30, 30)",
      "after": "whole original matrix determinant difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix determinant difference，不以小數近似替代。",
          "en": "Recompute whole original matrix determinant difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "30",
          "30"
        ],
        "output": "0",
        "label": "whole original matrix determinant difference"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "exact determinant=0;det-zero premise=是;SEPARATE full RREF rank=1",
      "after": "consequence=det_zero_only_excludes_full_rank;zero_determinant_does_not_determine_exact_rank",
      "relation": "implies",
      "rule_id": "rank_zero_determinant_not_exact_rank_or_failed_theorem",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零det只排除滿秩，零矩陣與非零singular矩陣可有不同rank；本題精確rank仍由RREF給。若det非零只是該前提不符，不是否定定理。",
        "en": "Zero determinant excludes full rank only.The zero matrix and a nonzero singular matrix can have different ranks;the exact rank here comes from RREF.A nonzero determinant makes the premise false,not the theorem."
      },
      "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,
      true,
      1,
      "det_zero_only_excludes_full_rank",
      "zero_determinant_does_not_determine_exact_rank"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 是, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)",
    "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": {
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}