If \(A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]\), then the inverse of the matrix \(A^3\) is

\(A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]\)
\[
\begin{aligned}
|A| & =-4+3=-1 \\
\operatorname{adj}(A) & =\left[\begin{array}{cc}
-2 & -3 \\
-(-1) & 2
\end{array}\right]^T=\left[\begin{array}{cc}
-2 & -3 \\
1 & 2
\end{array}\right]^T=\left[\begin{array}{cc}
-2 & 1 \\
-3 & 2
\end{array}\right] \\
A^{-1} & =\frac{\operatorname{adj}(A)}{|A|}=\frac{\left[\begin{array}{ll}
-2 & 1 \\
-3 & 2
\end{array}\right]}{(-1)}=\left[\begin{array}{cc}
2 & -1 \\
3 & -2
\end{array}\right]=A \\
\Rightarrow \quad A^{-1} & =A \Rightarrow A \cdot A^{-1}=A \cdot A \\
\Rightarrow \quad I & =A^2 \Rightarrow A \cdot I=A \cdot A^2 \\
\Rightarrow \quad A & =A^3 \Rightarrow(A)^{-1}=\left(A^3\right)^{-1} \\
\Rightarrow \quad A & =\left(A^3\right)^{-1} \Rightarrow\left(A^3\right)^{-1}=A
\end{aligned}
\]