# Cofactor Matrix Calculator

The calculator will find the matrix of cofactors of the given square matrix, with steps shown.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Find the cofactor matrix of $\left[\begin{array}{ccc}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{array}\right]$.

## Solution

The cofactor matrix consists of all cofactors of the given matrix, which are calculated according to the formula $C_{ij}=\left(-1\right)^{i+j}M_{ij}$, where $M_{ij}$ is the minor, i.e. the determinant of the submatrix formed by deleting row $i$ and column $j$ from the given matrix.

Calculate all cofactors:

$C_{11} = \left(-1\right)^{1 + 1} \left|\begin{array}{cc}5 & 6\\8 & 9\end{array}\right| = -3$ (for steps, see determinant calculator).

$C_{12} = \left(-1\right)^{1 + 2} \left|\begin{array}{cc}4 & 6\\7 & 9\end{array}\right| = 6$ (for steps, see determinant calculator).

$C_{13} = \left(-1\right)^{1 + 3} \left|\begin{array}{cc}4 & 5\\7 & 8\end{array}\right| = -3$ (for steps, see determinant calculator).

$C_{21} = \left(-1\right)^{2 + 1} \left|\begin{array}{cc}2 & 3\\8 & 9\end{array}\right| = 6$ (for steps, see determinant calculator).

$C_{22} = \left(-1\right)^{2 + 2} \left|\begin{array}{cc}1 & 3\\7 & 9\end{array}\right| = -12$ (for steps, see determinant calculator).

$C_{23} = \left(-1\right)^{2 + 3} \left|\begin{array}{cc}1 & 2\\7 & 8\end{array}\right| = 6$ (for steps, see determinant calculator).

$C_{31} = \left(-1\right)^{3 + 1} \left|\begin{array}{cc}2 & 3\\5 & 6\end{array}\right| = -3$ (for steps, see determinant calculator).

$C_{32} = \left(-1\right)^{3 + 2} \left|\begin{array}{cc}1 & 3\\4 & 6\end{array}\right| = 6$ (for steps, see determinant calculator).

$C_{33} = \left(-1\right)^{3 + 3} \left|\begin{array}{cc}1 & 2\\4 & 5\end{array}\right| = -3$ (for steps, see determinant calculator).

Thus, the cofactor matrix is $\left[\begin{array}{ccc}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{array}\right]$.

The cofactor matrix is $\left[\begin{array}{ccc}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{array}\right]$A.