If A= [ rr & - & ] and A+A^ =I, then the value of is - Vidyadip

MCQ

If $A=\left[\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ and $A+A^{\prime}=I,$ then the value of $\alpha$ is

A
$\frac{\pi}{6}$
✓
$\frac{\pi}{3}$
C
$\pi$
D
$\frac{3 \pi}{2}$

✓

Answer

Correct option: B.

$\frac{\pi}{3}$

We have $, A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right] $
and $ A+A^{\prime}=I $
$ \Rightarrow\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]+\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $
$\Rightarrow\left[\begin{array}{cc}2 \cos \alpha & 0 \\ 0 & 2 \cos \alpha\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $
$ \Rightarrow \quad 2 \cos \alpha=1 $
$\Rightarrow \cos \alpha=\frac{1}{2} $
$\Rightarrow \alpha=\frac{\pi}{3}$

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