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