Cofactor

Definition

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$, with $n \ge 2$. The cofactor of $A$ in position $(i, j)$ is $$C_{ij} = (-1)^{i+j} M_{ij},$$ where $M_{ij}$ is the first minor of $A$ that results when row $i$ and column $j$ are deleted.

Remarks

The signs $(-1)^{i+j}$ form a checkerboard pattern, with $+$ in the top left corner. The cofactors give the Laplace expansion of the determinant. The transpose of the matrix of cofactors is the adjugate matrix.

Explanations

L1 human

A cofactor is a minor with a sign. The sign is $+$ or $-$, according to a checkerboard pattern. For the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the cofactors are $C_{11} = d$, $C_{12} = -c$, $C_{21} = -b$ and $C_{22} = a$.