Answer
$$|A|=5, \quad \left|A^{-1}\right|= \frac{1}{5}.$$
Work Step by Step
Since $$A=\left[\begin{array}{rrr} {2} & {3} \\ {1} & {4}\end{array}\right],$$
then $$A^{-1}=\frac{1}{5}\left[\begin{array}{rrr} {4} & {-3} \\ {-1} & {2}\end{array}\right].$$
Now, we have
$$|A|=5, \quad \left|A^{-1}\right|=\frac{1}{5^2}(8-3)=\frac{1}{5}.$$
Hence, we verify that $$\left|A^{-1}\right|=\frac{1}{|A|}.$$
