{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $m \\le n$, let $A$ be an $m \\times n$ matrix and $B$ an $n \\times m$ matrix over $K$.\nFor a subset $S \\subseteq \\{1, \\dots, n\\}$ with $m$ elements, let $A_S$ be the $m \\times m$ matrix of the columns of $A$ in $S$.\nLet $B^S$ be the $m \\times m$ matrix of the rows of $B$ in $S$.\nThen\n$$\\det(AB) = \\sum_{S} \\det(A_S) \\det(B^S),$$\nwhere the sum runs over all subsets $S$ of $\\{1, \\dots, n\\}$ with $m$ elements.\n\n## Remarks\n\nFor $m = n$, the sum has one term, and the formula is $\\det(AB) = \\det A \\det B$.\nIf $m \u003e n$, then $\\det(AB) = 0$.\nFor $m = 1$, it is the formula for the product of a row and a column.\nThe determinants $\\det(A_S)$ and $\\det(B^S)$ are [[minors|0BT9BYWVRY]] of order $m$.\n\n## Explanations\n\n### L1 human\n\nThe Cauchy–Binet formula computes the determinant of a product of two rectangular matrices.\nIt sums products of matching minors of the two factors.\nIt generalizes the product rule for determinants of square matrices.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15A15"
    ],
    "name": "Cauchy–Binet formula",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q45286",
      "wikipedia": "Cauchy–Binet formula"
    },
    "sources": [
      {
        "locator": "Statement",
        "url": "https://en.wikipedia.org/wiki/Cauchy–Binet_formula"
      }
    ],
    "tag": "08WE2W7DYS",
    "uses": [
      "0C3CFHDN6R",
      "0SD528P0EK"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/08WE2W7DYS/",
  "links": [
    {
      "field": "uses",
      "target": "0C3CFHDN6R"
    },
    {
      "field": "uses",
      "target": "0SD528P0EK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/08WE2W7DYS.md",
  "msc": [
    "15A15"
  ],
  "name": "Cauchy–Binet formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/08W/08WE2W7DYS.md",
  "status": null,
  "tag": "08WE2W7DYS",
  "uri": "m2://08WE2W7DYS",
  "url": "https://math2.tamnd.com/wiki/cauchy-binet-formula/",
  "writing": {
    "tag": "08WE2W7DYS",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}