Determinant of an inverse

Computation. $AA^{-1}=I$, so
$$\det(A)\det(A^{-1}) = \det(AA^{-1})=\det I = 1.$$
Since $A$ is invertible, $\det A\neq0$, so $\det(A^{-1})=1/\det A$.

Conclusion. The determinant of the inverse is the inverse of the determinant.