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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $A$ be an invertible $n \\times n$ matrix over $K$, and let $u, v \\in K^n$ be column vectors.\nThen\n$$\\det(A + u v^T) = (1 + v^T A^{-1} u) \\det A,$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThe matrix $u v^T$ has rank at most $1$, so the lemma describes the effect of a rank one update on the determinant.\nIt follows from [[Sylvester's determinant theorem|0G24VQ1088]] with the matrices $A^{-1} u$ and $v^T$.\nA variant without inverses is $\\det(A + u v^T) = \\det A + v^T \\operatorname{adj}(A) u$.\n\n## Explanations\n\n### L1 human\n\nAdding a matrix of rank one, $u v^T$, changes the determinant by a simple factor.\nThe factor is $1 + v^T A^{-1} u$, a single number.\nSo we can update a determinant cheaply when we know the inverse.\n",
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