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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $A$ be an $n \\times n$ matrix over $K$, and let $c \\in K$.\nLet $i \\ne j$, and let $B$ be the matrix that results when $c$ times row $j$ of $A$ is added to row $i$.\nThen $\\det B = \\det A$, where $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nBy linearity in row $i$, $\\det B = \\det A + c \\det A'$, where $A'$ has two equal rows; so $\\det A' = 0$.\nThis is the reason why [[Gaussian elimination|0JVJ2WCXZH]] computes determinants efficiently.\nThe same holds for columns.\n\n## Explanations\n\n### L1 human\n\nAdding a multiple of one row to another row shears the box spanned by the rows.\nA shear does not change the volume.\nSo this row operation keeps the determinant.\n",
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