Cofactor (mathematics)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

In mathematics, a cofactor appears in the definition of the determinant of a square matrix.

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 minor

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

The adjugate matrix adj M (in older literature called adjoint matrix[1]) is the n×n matrix whose (i,j) entry is the (j,i) cofactor (note the transposition of the indices). Letting In be the n×n identity (unit) matrix, we have

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

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 non-zero, then M is invertible and its inverse is the following,

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

Example

Consider the following example matrix,

$M = \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{pmatrix}.$

Its minors are the determinants (vertical bars indicate a determinant):

$M_{11} = \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \\ \end{vmatrix}\quad M_{12} = \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \\ \end{vmatrix} \quad M_{13} = \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \\ \end{vmatrix} \quad M_{21} = \begin{vmatrix} a_2 & a_3 \\ c_2 & c_3 \\ \end{vmatrix} \quad M_{22} = \begin{vmatrix} a_1 & a_3 \\ c_1 & c_3 \\ \end{vmatrix} \quad$
$M_{23} = \begin{vmatrix} a_1 & a_2 \\ c_1 & c_2 \\ \end{vmatrix}\quad M_{31} = \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \\ \end{vmatrix} \quad M_{32} = \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \\ \end{vmatrix} \quad M_{33} = \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \\ \end{vmatrix} \quad$

The adjugate matrix of M is

$\mathrm{adj}M = A = \begin{pmatrix} M_{11} & -M_{21} & M_{31} \\ -M_{12} & M_{22} & -M_{32} \\ M_{13} & -M_{23} & M_{33} \\ \end{pmatrix},$

and the inverse matrix is

$M^{-1} = |M|^{-1} A\, .$

Indeed,

\begin{align} \left( M\; M^{-1}\right)_{11} & = |M|^{-1}\left( a_1 M_{11}- a_2 M_{12} + a_3 M_{13}\right) = \frac{|M|}{|M|} = 1 \\ \left( M\; M^{-1}\right)_{21} & = |M|^{-1}\left( b_1 M_{11}- b_2 M_{12} + b_3 M_{13}\right) =|M|^{-1}\left[ b_1(b_2c_3-b_3c_2) - b_2(b_1c_3-b_3c_1) + b_3(b_1c_2-b_2c_1)\right] = 0 ,\\ \end{align}

and the other matrix elements of the product follow likewise.

Note

1. The term "adjoint" for the adjugate matrix is disappearing because it is felt that it is easily confused with Hermitian adjoint, the transpose and complex conjugate of a matrix.