Proof

Let \(E\) be a identity matrix:

$$E=\begin{bmatrix}1 &0&\cdots&0\\0 &1&\cdots&0\\\vdots &\vdots & \ddots &\vdots\\0&0&\cdots&1\\\end{bmatrix}$$

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

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

Therefore, we obtain

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