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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $n \\ge 2$, and let $A$ be an $n \\times n$ matrix over $K$.\nThen\n$$A \\operatorname{adj}(A) = \\operatorname{adj}(A) A = \\det(A)\\, I_n,$$\nwhere $\\operatorname{adj}(A)$ is the [[adjugate|07W9H8QF38]] of $A$ and $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThe diagonal entries of $A \\operatorname{adj}(A)$ are the [[Laplace expansions|0C08BQ3M05]] of $\\det A$ along the rows.\nThe entries off the diagonal are expansions of matrices with two equal rows, so they are zero.\nThe identity holds for all square matrices, also for singular ones.\n\n## Explanations\n\n### L1 human\n\nMultiplying a matrix by its adjugate gives the determinant times the identity matrix.\nSo, if the determinant is invertible, then dividing the adjugate by it gives the inverse.\nFor a singular matrix, the product is the zero matrix.\n",
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