Question

If $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]$, then $(A+B)(A-B)$ is equal to

• A. $A^2-B^2$
• B. $A^2+B^2$
• C. $A^2-B^2+BA+AB$
• D. None of these

Answer 1

Answer: (A)

$A^2-B^2$

Given $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]$
$AB=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
$BA=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
$\Rightarrow BA=AB$
Now, $(A+B)(A-B)=A^2-AB+BA-B^2$
$\Rightarrow (A+B)(A-B)=A^2-B^2$