{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R194",
  "content_hash": null,
  "slug": "dbm-claim-exact-determinant",
  "type": "claim",
  "title": "Exact determinant of the disjoint-binary-support matrix",
  "summary": "The determinant is zero at odd orders and \\((-1)^{n/2}\\) at even orders.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "For \\(1\\leq i,j\\leq n\\), let \\((A_n)_{ij}=1\\) when \\(i\\mathbin{\\&}j=0\\). Then\n\\[\n\\det A_n=\\begin{cases}0,&n\\text{ odd},\\\\(-1)^{n/2},&n\\text{ even}.\\end{cases}\n\\]\nFor even order \\(n=2m\\), place odd indices before even indices. The matrix becomes\n\\[\n\\begin{pmatrix}0&C_m\\\\C_m^T&A_m\\end{pmatrix},\n\\]\nwhere \\(C_m=(\\mathbf1_{i\\mathbin{\\&}j=0})_{0\\leq i<m,\\,1\\leq j\\leq m}\\). Swapping the two column blocks gives a block triangular matrix and contributes sign \\((-1)^{m^2}=(-1)^m\\). Since \\(C_m\\) is unimodular, \\(\\det A_{2m}=(-1)^m\\).\n\nAt odd order \\(2m+1\\), the same split has an \\((m+1)\\times m\\) off-diagonal block. Its transpose has a nonzero kernel vector, which yields a kernel vector of \\(A_{2m+1}\\). Thus the odd determinant vanishes.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every positive matrix order n"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2210.12913",
      "locator": "Mereb, On determinants of matrices related to Pascal's triangle, Corollary 1, combined with the parity block decomposition recorded here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2210.12913",
    "locator": "Mereb, On determinants of matrices related to Pascal's triangle, Corollary 1, combined with the parity block decomposition recorded here"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R195",
      "title": "The required Pascal blocks are unimodular",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R196",
      "title": "Rank and Smith normal form follow from the determinant formula",
      "object_type": "claim",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "R193",
      "title": "Exact determinant sweep through order 130",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R1277",
      "title": "Lean proof of the exact disjoint-support determinant formula",
      "object_type": "formalization",
      "relation": "formalizes",
      "direction": "incoming"
    },
    {
      "slug": "disjoint-binary-support-determinant",
      "title": "disjoint binary support determinant",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}