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Cofactor matrix of [1234]\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]

The calculator will find the matrix of cofactors of the square 22x22 matrix [1234]\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right], with steps shown.
A

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Your Input

Find the cofactor matrix of [1234]\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right].

Solution

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

Calculate all cofactors:

C11=(−1)1+1∣4∣=4C_{11} = \left(-1\right)^{1 + 1} \left|\begin{array}{c}4\end{array}\right| = 4 (for steps, see determinant calculator).

C12=(−1)1+2∣3∣=−3C_{12} = \left(-1\right)^{1 + 2} \left|\begin{array}{c}3\end{array}\right| = -3 (for steps, see determinant calculator).

C21=(−1)2+1∣2∣=−2C_{21} = \left(-1\right)^{2 + 1} \left|\begin{array}{c}2\end{array}\right| = -2 (for steps, see determinant calculator).

C22=(−1)2+2∣1∣=1C_{22} = \left(-1\right)^{2 + 2} \left|\begin{array}{c}1\end{array}\right| = 1 (for steps, see determinant calculator).

Thus, the cofactor matrix is [4−3−21]\left[\begin{array}{cc}4 & -3\\-2 & 1\end{array}\right].

Answer

The cofactor matrix is [4−3−21]\left[\begin{array}{cc}4 & -3\\-2 & 1\end{array}\right]A.