Question

If $A=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$, $B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, $C=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ then which of the following statement is true

• A. $AB=BA$
• B. $AB=-BA$
• C. $A^2=BC$
• D. $A^2=B+C$

Answer 1

Answer: (B)

$AB=-BA$

For $A=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
$AB=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}-i& 0\\ 0& i\end{array}\right]BA=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]=\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right]=-AB{A}^2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]BC=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]B+C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\therefore AB=-BA$