Question

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

• A. $A$
• B. $1$
• C. $-1$
• D. $-A$

Answer 1

Answer: (A)

$A=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$
$A^{-1}=\frac{1}{-1}\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]=A$
$A^2=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$
$=\left[\begin{array}{cc}4-3& -2+2\\ 6-6& -3+4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$
$\therefore A^3=A$