Question

Assertion :If $A$ and $B$ are two $3\times 3$ matrices such that $AB=0$, then $A=0$ or $B=0$. Reason: If $A$, $B$ and $X$ are three $3\times 3$ matrices such that $AX=B$, $|A|\ne 0$, then $X=A^{-1}B$.

• 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 D Assertion is incorrect but Reason is correct

Statement-2 is true since $|A|\ne 0$, implies $A^{-1}$ exists.
$\therefore AX=B$
$\Rightarrow A^{-1}\left(AX\right)=A^{-1}B$
$\Rightarrow \left(A^{-1}A\right)X=A^{-1}B$
$\Rightarrow IX=A^{-1}B$
$\Rightarrow X=A^{-1}B$
That statement-1 is false can be seen by the following example.
Let $A=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]thenAB=0$
but neither $A=0$ nor $B=0$.
$\therefore$ Assertion is incorrect but Reason is correct.