Question

If $A=\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right],B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],C=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$, then $A^2+B^2+C^2=$

• A. $2I$
• B. $-2I$
• C. $-3I$
• D. $3I$

Answer 1

Answer: (C)

$-3I$

$A=\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right],B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],C=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$
$A^2+B^2+C^2$
$=\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right]+\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]+\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$
$=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$=\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right]=-3I$