Let M be a square matrix of size n. The (i,j) minor refers to the determinant of the (n-1)×(n-1) submatrix Mi,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix Mi,j itself). The corresponding cofactor is the signed determinant
The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have
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
Consider the following example matrix,
Its minors are the determinants (bars indicate a determinant):
The adjugate matrix of M is
and the inverse matrix is
and the other matrix elements of the product follow likewise.