Answer
$$\begin{pmatrix}-\frac{1}{10}&\frac{1}{5}\\ \frac{2}{5}&\frac{1}{5}\end{pmatrix}$$
Work Step by Step
We know the following equation for the inverse of a matrix:
$$\begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}^{-1}=\frac{1}{\det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}}\begin{pmatrix}d\:&\:-b\:\\ -c\:&\:a\:\end{pmatrix}$$
Thus, we obtain:
$$\frac{1}{\det \begin{pmatrix}-2&2\\ 4&1\end{pmatrix}}\begin{pmatrix}1&-2\\ -4&-2\end{pmatrix} \\ \begin{pmatrix}-\frac{1}{10}&\frac{1}{5}\\ \frac{2}{5}&\frac{1}{5}\end{pmatrix}$$
