Answer
The statement does not hold for the matrix $A=\left[\begin{array}{cc} 2 & 2\\ 4 & 4\end{array}\right]$ for example.
Work Step by Step
Consider the following statement:
If $A$ is a matrix, then $\det\left(A^{-1}\right)=\displaystyle\frac{1}{\det A}$.
We will now present a counterexample in order to show that the statement is false.
Let $A=\left[\begin{array}{cc} 2 & 2\\ 4 & 4\end{array}\right]$. We have that $\det A=0$ which implies that $A$ is not invertible and therefore, $A^{-1}$ does not exist.
Thus, the equation $\det\left(A^{-1}\right)=\displaystyle\frac{1}{\det A}$ is not even well-defined for $A$ and therefore, the statement is false.
$\textbf{Observation:}$ If in the statement $A$ were required to be invertible then, the statement would be true by a result of Chapter $3$.
