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  "body": "\n## Proof\n\n1. Let $r_k$ be row $k$ of $A$ for $k \\ne i$. By hypothesis, $r_k$ is also row $k$ of $B$ and of $C$.\n2. By property 1 of the [[definition of the determinant|0C3CFHDN6R]], $\\det$ is a [[multilinear function|0443XNARDV]] of the $n$ rows.\n3. For a row $x \\in K^n$, let $M(x)$ have row $i$ equal to $x$ and row $k$ equal to $r_k$ for $k \\ne i$. Put $\\varphi(x) = \\det M(x)$. By step 2, $\\varphi$ is linear.\n4. Linearity of $\\varphi$ means $\\varphi(sa + tb) = s \\varphi(a) + t \\varphi(b)$ for all rows $a, b$ and all $s, t \\in K$.\n5. By step 1, $\\varphi(a) = \\det A$, $\\varphi(b) = \\det B$ and $\\varphi(c) = \\det C$.\n6. Since $c = sa + tb$, step 4 gives $\\det C = \\varphi(sa + tb) = s \\det A + t \\det B$.\n\n## Idea\n\nThis is property 1 of the definition, unpacked. Multilinear means linear in each row when the other rows are held fixed. The three matrices share all rows except row $i$, so linearity in row $i$ is exactly the claim.\n",
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