Answer
$$|A|=24, \quad \left|A^{-1}\right|=\frac{1}{24}.$$
Work Step by Step
Since $$A=\left[\begin{array}{rrr} 1&0&-1&3\\ 1&0&3&-2
\\ 2&0&2&-1\\ 1&-3&1&-2
\end{array}\right],$$
then one can calculate its inverse , that is,
$$A^{-1}= \left[\begin{array}{rrr} -\frac{1}{8}&-\frac{5}{8}&{\frac {7}{8}}&0
\\ -\frac{1}{4}&-\frac{1}{4}&{\frac {5}{12}}&-\frac{1}{3}
\\ \frac{3}{8}&{\frac {7}{8}}&-\frac{5}{8}&0\\ \frac{1}{2}
&\frac{1}{2}&- \frac{1}{2}&0
\end{array}\right].$$
Now, we have
$$|A|=24, \quad \left|A^{-1}\right|=\frac{1}{24}.$$
Hence, we verify that $$\left|A^{-1}\right|=\frac{1}{|A|}.$$
