Answer
$$|A|=6, \quad \left|A^{-1}\right|= \frac{1}{6}.$$
Work Step by Step
Since $$A=\left[\begin{array}{rrr} {1} & {-2} \\ {2} & {2}\end{array}\right],$$
then $$A^{-1}=\frac{1}{6}\left[\begin{array}{rrr} {2} & {2} \\ {-2} & {1}\end{array}\right].$$
Now, we have
$$|A|=6, \quad \left|A^{-1}\right|=\frac{1}{6^2}(2+4)=\frac{1}{6}.$$
Hence, we verify that $$\left|A^{-1}\right|=\frac{1}{|A|}.$$
