Question

Assertion :Let $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $B=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$

If $b=0$, then $AB=BA$

Reason: If $b\ne 0$, then $AB=BA$
$\iff$ $p=sandq=r$.

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

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $B=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$

Assertion:
If $b=0,$ then $AB=BA$

$AB=\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]=\left[\begin{array}{cc}ap& aq\\ ar& as\end{array}\right]$

$BA=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]=\left[\begin{array}{cc}ap& aq\\ ar& as\end{array}\right]$

$\therefore$ Assertion is true.

Reason:
If $b\ne 0$ then $AB=BA$ if and only if $p=s$ and $q=r$

equating $AB$ and $BA$ gives
$ap+br=pa+qb\Rightarrow q=r$ and

$aq+bs=pb+aq\Rightarrow p=s$

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.