Let A = begin{bmatrix} cos alpha & -sin alpha | vedclass

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(D) Given $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \ \sin \alpha & \cos \alpha \end{bmatrix}$.By the property of rotation matrices,$A^n = \beg

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$\frac{\pi}{64}$

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(D) Given $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \ \sin \alpha & \cos \alpha \end{bmatrix}$.By the property of rotation matrices,$A^n = \begin{bmatrix} \cos(n\alpha) & -\sin(n\alpha) \ \sin(n\alpha) & \cos(n\alpha) \end{bmatrix}$.Thus,$A^{32} = \begin{bmatrix} \cos(32\alpha) & -\sin(32\alpha) \ \sin(32\alpha) & \cos(32\alpha) \end{bmatrix}$.We are given $A^{32} = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}$.Comparing the elements,we get $\cos(32\alpha) = 0$ and $\sin(32\alpha) = 1$.This implies $32\alpha = 2n\pi + \frac{\pi}{2}$ for some integer $n$.For $n=0$,$32\alpha = \frac{\pi}{2}$,which gives $\alpha = \frac{\pi}{64}$.

List-$I$	List-$II$
a) $A$ matrix which is not a square matrix	i) Symmetric matrix
b) $A$ square matrix $A' = A$	ii) Null matrix
c) The diagonal elements of a diagonal matrix are same	iii) Rectangular matrix
d) $A$ matrix which is both symmetric and skew symmetric	iv) Scalar matrix

Let $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$,$\alpha \in R$ such that $A^{32} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Then a value of $\alpha$ is

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