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  "body": "\n## Statement\n\nLet $A$ be a square matrix, and let $A'$ be a matrix obtained from $A$ by one elementary [[row operation|04XWH28452]]. Write $\\det$ for the [[determinant|00KDZRW4C0]].\n\nIf the operation multiplies one row by a scalar $k$, then $\\det(A') = k\\det(A)$.\n\nIf the operation swaps two rows, then $\\det(A') = -\\det(A)$.\n\nIf the operation adds a multiple of one row to another row, then $\\det(A') = \\det(A)$.\n\n## Remarks\n\nThe three cases give a way to compute a determinant by row reduction. One tracks the scalings and the swaps along the way.\n",
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