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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $n \\ge 1$, and let $1 \\le i \\le n$.\nLet $A$, $B$, $C$ be $n \\times n$ matrices over $K$ that agree outside row $i$.\nLet $a$, $b$, $c$ be row $i$ of $A$, $B$, $C$, and assume $c = sa + tb$ with $s, t \\in K$.\nThen\n$$\\det C = s \\det A + t \\det B,$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThis is property 1 of the definition of the determinant.\nThe determinant is not additive in the whole matrix: in general $\\det(A + B) \\ne \\det A + \\det B$.\nBy $\\det A^T = \\det A$, the determinant is also multilinear in the columns.\n\n## Explanations\n\n### L1 human\n\nIf we fix all rows but one, then the determinant is a linear function of the remaining row.\nSo a sum in one row splits the determinant into a sum of two determinants.\nA factor in one row comes out of the determinant.\n",
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