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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $A = (a_{ij})$ be an $n \\times n$ matrix over $K$, and let $\\sigma$ be a permutation of $\\{1, \\dots, n\\}$.\nLet $B = (b_{ij})$ be the matrix with $b_{ij} = a_{\\sigma(i), j}$.\nThen $\\det B = \\sgn(\\sigma) \\det A$, where $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nIn particular, exchanging two rows changes the sign of the determinant.\nFor a [[permutation matrix|08SP3DF8NB]] $P$, this gives $\\det P = \\pm 1$.\nThe same holds for a permutation of the columns.\n\n## Explanations\n\n### L1 human\n\nWhen we exchange two rows, the determinant changes its sign.\nA general reordering of the rows is a sequence of such exchanges.\nSo it multiplies the determinant by $1$ or by $-1$, according to the sign of the permutation.\n",
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