Question

If $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, then $A^8$ equals

• A. $4B$
• B. $128B$
• C. $-128B$
• D. $-64B$

Answer 1

The correct option is B $128B$

Given, $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$
$=i\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
$=iB$
$\Rightarrow A=iB$
$\Rightarrow A^2=-B^2$
Now, $B^2=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
$B^2=\left[\begin{array}{cc}2& -2\\ -2& 2\end{array}\right]=2B$
$\Rightarrow A^2=2B$
$\Rightarrow A^4=4B^2=8B$
$\to A^8=64B^2=128B$