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Cofactor matrix of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$
Your Input
Find the cofactor matrix of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$.
Solution
The cofactor matrix consists of all cofactors of the given matrix, which are calculated according to the formula $$$C_{ij}=\left(-1\right)^{i+j}M_{ij}$$$, 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:
$$$C_{11} = \left(-1\right)^{1 + 1} \left|\begin{array}{c}4\end{array}\right| = 4$$$ (for steps, see determinant calculator).
$$$C_{12} = \left(-1\right)^{1 + 2} \left|\begin{array}{c}3\end{array}\right| = -3$$$ (for steps, see determinant calculator).
$$$C_{21} = \left(-1\right)^{2 + 1} \left|\begin{array}{c}2\end{array}\right| = -2$$$ (for steps, see determinant calculator).
$$$C_{22} = \left(-1\right)^{2 + 2} \left|\begin{array}{c}1\end{array}\right| = 1$$$ (for steps, see determinant calculator).
Thus, the cofactor matrix is $$$\left[\begin{array}{cc}4 & -3\\-2 & 1\end{array}\right]$$$.
Answer
The cofactor matrix is $$$\left[\begin{array}{cc}4 & -3\\-2 & 1\end{array}\right]$$$A.