Question

if P =$\left[\begin{array}{ccc}i& 0& -i\\ 0& -i& i\\ -i& i& 0\end{array}\right]$ and $Q=\left[\begin{array}{cc}-i& i\\ 0& 0\\ i& -i\end{array}\right]$ then $PQ$ is equal to

• A. $\left[\begin{array}{cc}-2& 2\\ 1& -1\\ 1& -1\end{array}\right]$
• B. $\left[\begin{array}{cc}2& -2\\ -1& 1\\ -1& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}2& -2\\ -1& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Answer 1

Answer: (B)

Since, $P=\left[\begin{array}{ccc}i& 1& -i\\ 0& -i& i\\ -i& i& 0\end{array}\right]andQ=\left[\begin{array}{cc}-i& i\\ 0& 0\\ i& -i\end{array}\right]$

$\therefore PQ=\left[\begin{array}{ccc}i& 0& -i\\ 0& -i& i\\ -i& i& 0\end{array}\right]\left[\begin{array}{cc}-i& i\\ 0& 0\\ i& -i\end{array}\right]$

$=\left[\begin{array}{cc}-i^2-i^2& i^2+i^2\\ i^2& -i^2\\ i^2& -i^2\end{array}\right]\left[\begin{array}{cc}1+1& -1-1\\ -1& 1\\ -1& 1\end{array}\right]=\left[\begin{array}{cc}2& -2\\ -1& 1\\ -1& 1\end{array}\right]$