Cofactor (mathematics)

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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 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

(-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 . \,

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