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  "body": "\n## Definition\n\nLet $K$ be a commutative ring and let $A$ be an $n \\times n$ matrix over $K$, with $n \\ge 2$.\nThe *cofactor* of $A$ in position $(i, j)$ is\n$$C_{ij} = (-1)^{i+j} M_{ij},$$\nwhere $M_{ij}$ is a first [[minor|0BT9BYWVRY]] of $A$. It is the minor that results when row $i$ and column $j$ are deleted.\n\n## Remarks\n\nThe signs $(-1)^{i+j}$ form a checkerboard pattern, with $+$ in the top left corner.\nThe cofactors give the [[Laplace expansion|0C08BQ3M05]] of the determinant.\nThe transpose of the matrix of cofactors is the [[adjugate matrix|07W9H8QF38]].\n\n## Explanations\n\n### L1 human\n\nA cofactor is a minor with a sign.\nThe sign is $+$ or $-$, according to a checkerboard pattern.\nFor the matrix $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$, the cofactors are $C_{11} = d$, $C_{12} = -c$, $C_{21} = -b$ and $C_{22} = a$.\n",
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