Answer
$$A^{-1}=\frac{1}{8}\left[\begin{array}{ccc}{2} &{2} \\{-2}& {2} \end{array}\right].$$
Work Step by Step
Let the matrix $A$ be given by
$$A=\left[\begin{array}{ccc}{2} &{-2} \\{2}& {2} \end{array}\right].$$
Since $ad-bc=4+4=8\neq 0$, then $A$ has inverse given by
$$A^{-1}=\frac{1}{8}\left[\begin{array}{ccc}{2} &{2} \\{-2}& {2} \end{array}\right].$$