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  "body": "\n## Proof\n\n1. Let $A$ be an $n \\times n$ matrix over $K$ with rows $i \\ne j$ that are equal.\n2. By property 2 of the [[definition of the determinant|0C3CFHDN6R]], $\\det$ vanishes on every matrix with two equal rows.\n3. So $\\det A = 0$.\n\n## Idea\n\nIn this course the determinant is defined as the unique alternating multilinear function of the rows with value $1$ on the identity. Vanishing on two equal rows is the alternating property, so the claim is part of the definition.\n",
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