Cofactor Matrix Calculator

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

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Your Input

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 $$$\left[\begin{array}{ccc}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{array}\right]$$$.

Answer

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