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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 $c \\in K$.\nThen\n$$\\det(cA) = c^n \\det A,$$\nwhere $cA$ is the [[scalar multiple|09V6DDACXR]] and $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nEach of the $n$ rows is multiplied by $c$, and each row contributes one factor $c$.\nFor example, $\\det(-A) = (-1)^n \\det A$.\nSo an $n \\times n$ real [[skew-symmetric matrix|0PBY4E9GQF]] with $n$ odd has determinant $0$.\n\n## Explanations\n\n### L1 human\n\nScaling a matrix by $c$ stretches every edge of a box by $c$.\nIn $n$ dimensions, this multiplies the volume by $c^n$.\nSo doubling a $3 \\times 3$ matrix multiplies its determinant by $8$.\n",
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