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 M_i,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

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