{
  "aliases": [
    "cofactor-expansion"
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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $n \\ge 2$, and let $A = (a_{ij})$ be an $n \\times n$ matrix over $K$.\nLet $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.\nThen, for each row $i$,\n$$\\det A = \\sum_{j=1}^{n} a_{ij} C_{ij} = \\sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$\n\n## Remarks\n\nThe expansion reduces a determinant of order $n$ to $n$ determinants of order $n - 1$.\nIt is efficient when the chosen row contains many zeros.\nThere is a similar [[expansion along a column|06AMCJMMVM]].\nReplace the entries of row $i$ by those of another row $k$.\nThen the sum is $0$, because the new matrix has two equal rows.\n\n## Explanations\n\n### L1 human\n\nTo expand a determinant along a row, take each entry of that row.\nMultiply it by its cofactor, which is the signed determinant of what is left after deleting its row and column.\nAdd these products.\n",
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  "name": "Laplace expansion along a row",
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