Question

If $A^3=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]\&A^4=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ where $i^2=-1$ then $A^2=$ _________

• A. $\left[\begin{array}{cc}-i& -0\\ 0& -i\end{array}\right]$
• B. $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
• C. $\left[\begin{array}{cc}0& -1\\ -i& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$
Here, $|A^3|=1$