Answer
$$|A|=-2, \quad \left|A^{-1}\right|=-\frac{1}{2}.$$
Work Step by Step
Since $$A=\left[\begin{array}{rrr} 1&0&2\\ 2&-1&3
\\ 1&-2&2
\end{array}\right],$$
then one can calculate its inverse , that is,
$$A^{-1}= \left[\begin{array}{rrr} -2&2&-1\\ \frac{1}{2}&0&-\frac{1}{2}
\\ \frac{2}{2}&-1&\frac{1}{2}
\end{array}\right].$$
Now, we have
$$|A|=-2, \quad \left|A^{-1}\right|=-\frac{1}{2}.$$
Hence, we verify that $$\left|A^{-1}\right|=\frac{1}{|A|}.$$
