Question

Assertion :If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ \cos \alpha & \sin \alpha \end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \alpha & \cos \alpha \\ \sin \alpha & \sin \alpha \end{array}\right]$, then $AB=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$. Reason: The product of two matrices can never be equal to an unit matrix.

• 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. Both assertion and reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ \cos \alpha & \sin \alpha \end{array}\right]$ $B=\left[\begin{array}{cc}\cos \alpha & \cos \alpha \\ \sin \alpha & \sin \alpha \end{array}\right]$

Applying matrix multiplication

$AB=\left[\begin{array}{cc}{\cos }^2\alpha +{\sin }^2\alpha & {\cos }^2\alpha +{\sin }^2\alpha \\ {\cos }^2\alpha +{\sin }^2\alpha & {\cos }^2\alpha +{\sin }^2\alpha \end{array}\right]$

$=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

Therefore, the assertion is correct.

And, the reason is incorrect as for given $A,B$ we get $AB$ is always a unit matrix.

So, option C is correct