Proof

Let $E$ be a identity matrix:

$$E=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]$$

Then, by the property of determinant (8), we have

$$1=|E|=|A{A}^{-1}|=|A||A^{-1}|$$

Therefore, we obtain

$$|A^{-1}|=\frac{1}{|A|}$$