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  "body": "\n## Proof\n\n1. Let $a_k$ be row $k$ of $A$. Then row $i$ of $B$ is $a_i + c a_j$, and the other rows of $B$ are those of $A$.\n2. Let $A'$ be the matrix obtained from $A$ by replacing row $i$ with $a_j$.\n3. The matrices $A$, $A'$ and $B$ agree outside row $i$. Their rows $i$ are $a_i$, $a_j$ and $1 \\cdot a_i + c \\cdot a_j$.\n4. By [[multilinearity in the rows|0PPG793642]] with $s = 1$ and $t = c$, we get $\\det B = \\det A + c \\det A'$.\n5. Rows $i$ and $j$ of $A'$ are both $a_j$, and $i \\ne j$. So $\\det A' = 0$, by the rule for [[two equal rows|0K4VQB7HWR]].\n6. By steps 4 and 5, $\\det B = \\det A$.\n\n## Idea\n\nSplit the new row into its old part and the added part. The added part gives a matrix with a repeated row, and that determinant is zero.\n",
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  "name": "Proof that adding a multiple of one row to another preserves the determinant",
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