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  "body": "\n## Proof\n\n1. Let $i$ be the row of $A$ that is multiplied by $c$, and let $a$ be row $i$ of $A$. Then row $i$ of $B$ is $ca$.\n2. The matrices $A$, $A$ and $B$ agree outside row $i$. Their rows $i$ are $a$, $a$ and $ca = c \\cdot a + 0 \\cdot a$.\n3. By [[multilinearity in the rows|0PPG793642]] with $s = c$ and $t = 0$, we get $\\det B = c \\det A + 0 \\cdot \\det A$.\n4. So $\\det B = c \\det A$.\n\n## Idea\n\nLinearity in one row includes homogeneity: a scalar factor in that row comes out of the determinant. We get it from the linearity rule by taking the second coefficient to be zero.\n",
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