Question

If $A=\begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}$, then the inverse of the matrix $A^{3}$ is

• A. $A$
• B. $1$
• C. $-1$
• D. $- A$

Answer 1

Answer: (A)

$A=\begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}$
$A^{-1}=\frac{1}{-1}\begin{bmatrix}-2 & 1 \\ -3 & 2\end{bmatrix}=\begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}=A$
$A^{2}=\begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}\begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}$
$=\begin{bmatrix}4-3 & -2+2 \\ 6-6 & -3+4\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}=I$
$\therefore A^{3}=A$