if A =

Question:

If $\mathrm{A}=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)$ and $\operatorname{det}\left(\mathrm{A}^{2}-\frac{1}{2} \mathrm{I}\right)=0$, then a possible value of $\alpha$ is

  1. (1) $\frac{\pi}{2}$

  2. (2) $\frac{\pi}{3}$

  3. (3) $\frac{\pi}{4}$

  4. (4) $\frac{\pi}{6}$


Correct Option: , 3

Solution:

$A^{2}=\sin ^{2} \alpha I$

So, $\left|A^{2}-\frac{1}{2}\right|=\left(\sin ^{2} \alpha-\frac{1}{2}\right)^{2}=0$

$\Rightarrow \sin \alpha=\pm \frac{1}{\sqrt{2}}$

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