Cofactor Matrix

The cofactor matrix of a square matrix $ M = [a_{i,j}] $ is noted $ Cof(M) $. It is the matrix of the cofactors, i.e. the minors weighted by a factor $ (-1)^{i+j} $.

For each item in the matrix, compute the determinant of the sub-matrix $ SM $ associated. The determinant is noted $ \text{Det}(SM) $ or $ | SM | $ and is also called minor. To calculate $ Cof(M) $ multiply each minor by a $ -1 $ factor according to the position in the matrix.

$$ Cof_{i,j} = (-1)^{i+j} \text{Det}(SM_i) $$

Calculation of a 2x2 cofactor matrix:

$$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

$$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

$$ Cof(M) = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix} $$

Calculation of a 3x3 cofactor matrix:

$$ M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} $$

$$ Cof(M) = \begin{bmatrix} + \begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$

The transpose of the cofactor matrix (comatrix) is the adjoint matrix.

Most of the properties of the cofactor matrix actually concern its transpose, the transpose of the matrix of the cofactors is called adjugate matrix.

$$ A({}^t{{\rm com} A}) = ({}^t{{\rm com} A})A =\det{A} \times I_n $$

$$ A^{-1}=\frac1{\det A} \, {}^t{{\rm com} A} $$

A cofactor is calculated from the minor of the submatrix.

$$ Cof_{i,j} = (-1)^{i+j} \text{Det}(SM_i) $$