Cofactor Matrix

linear-algebra

Definition

Cofactor Matrix

Let $A=(a_{ij})\in K^{n\times n}$ be a square matrix over a field $K$. The cofactor matrix of $A$ is the matrix

$$C=(c_{ij})\in K^{n\times n}$$

whose entries are the cofactors

$$c_{ij}=(-1{)}^{i+j}\det (A_{ij}),$$

where $A_{ij}$ is the submatrix obtained from $A$ by deleting the $i$-th row and the $j$-th column.

The adjugate matrix is the transpose of the cofactor matrix:

$$\mathrm{adj}(A)=C^T.$$

Example: For

$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),$$

the cofactor matrix is

$$C=\left(\begin{array}{cc}d& -c\\ -b& a\end{array}\right).$$