Question

Assertion :Let $A=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)$,$B=\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)$,where $i=\sqrt{-1}$

then $(A+B)(A-B)$ equals $A^2-B^2$

Reason: $AB=BA$

• A. Both Assertion and Reason are correct and the Reason is the correct explanation for the Assertion.
• B. Both Assertion and Reason are correct but the Reason is Not the correct explanation for the Assertion.
• C. Assertion is correct while the Reason is incorrect.
• D. Assertion is incorrect while the Reason is correct.

Answer 1

The correct option is A Both Assertion and Reason are correct and the Reason is the correct explanation for the Assertion.

$AB=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

and $BA=\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ $AB=BA$

$\therefore$ Reason (R) is true.

Now $(A+B)(A-B)=A^2-AB+BA-B^2=A^2-B^2$
Thus both the Assertion and Reason are correct and the Reason is the correct explanation for the Assertion.