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  "body": "\n## Definition\n\nLet $K$ be a commutative ring, let $n \\ge 2$, and let $A$ be an $n \\times n$ matrix over $K$ with [[cofactors|0XPKR18APA]] $C_{ij}$.\nThe *adjugate* of $A$ is the $n \\times n$ matrix $\\operatorname{adj}(A)$ with entries\n$$\\operatorname{adj}(A)_{ij} = C_{ji}.$$\nSo $\\operatorname{adj}(A)$ is the [[transpose|09DXWYRD43]] of the matrix of cofactors $(C_{ij})$.\n\n## Remarks\n\nFor $n = 2$, $\\operatorname{adj} \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} = \\begin{pmatrix} d \u0026 -b \\\\ -c \u0026 a \\end{pmatrix}$.\nThe adjugate satisfies [[$A \\operatorname{adj}(A) = \\det(A) I_n$|0528HKWZ1Z]].\nOlder texts call it the *classical adjoint*; it is not the adjoint of an inner product space.\n\n## Explanations\n\n### L1 human\n\nTo build the adjugate, replace each entry by its cofactor, and then transpose.\nThe adjugate is almost the inverse: it is the inverse multiplied by the determinant.\nFor $2 \\times 2$ matrices, swap the diagonal entries and negate the other two.\n",
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