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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $A$ be an $m \\times n$ matrix and $B$ an $n \\times m$ matrix over $K$.\nThen\n$$\\det(I_m + AB) = \\det(I_n + BA),$$\nwhere $I_m$ and $I_n$ are [[identity matrices|06DW9BAWB1]] and $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThe two sides are determinants of different sizes, $m$ and $n$.\nFor a column $u$ and a row $v^T$, it gives $\\det(I_m + u v^T) = 1 + v^T u$.\nThe statement is also called the Weinstein–Aronszajn identity.\n\n## Explanations\n\n### L1 human\n\nThe products $AB$ and $BA$ can have very different sizes.\nStill, adding the identity to them gives the same determinant.\nThis can turn a large determinant into a small one.\n",
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