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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 column $j$,\n$$\\det A = \\sum_{i=1}^{n} a_{ij} C_{ij} = \\sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$\n\n## Remarks\n\nIt follows from the [[expansion along a row|0C08BQ3M05]], applied to $A^T$, because $\\det A^T = \\det A$.\nFor example, expansion along column $1$ of $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 0 \u0026 4 \u0026 5 \\\\ 0 \u0026 6 \u0026 7 \\end{pmatrix}$ gives $1 \\cdot (28 - 30) = -2$.\n\n## Explanations\n\n### L1 human\n\nThe same expansion works down a column instead of across a row.\nWe choose the column with the most zeros, to save work.\nEach zero entry removes one smaller determinant from the sum.\n",
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