Cofactor (mathematics)

In mathematics, a cofactor is a component of a matrix computation of the matrix determinant.

Let M be a square matrix of size n. The (i,j) minor is the (n-1)×(n-1) submatrix Mi,j formed by deleting the i-th row and j-th column from M. The corresponding cofactor is the signed determinant


 * $$(-1)^{i+j} \det M_{i,j} . \,$$

The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have


 * $$M \cdot \mathop{\mbox{adj}} M = (\det M) I_n = \mathop{\mbox{adj}} M \cdot M ,\,$$

which encodes the rule for expansion of the determinant of M by any the cofactors of any row or column. This expression shows that if det M is invertible, then M is invertible and the matrix inverse is determined as


 * $$M^{-1} = (\det M)^{-1} \mathop{\mbox{adj}} M . \,$$