Question

Let \(A = \begin{pmatrix}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha\end{pmatrix}\), \((\alpha \in R)\) such that \(A^{32} = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\) then a value of \(\alpha\) is

• A. \(0\)
• B. \(\frac{\pi}{16}\)
• C. \(\frac{\pi}{32}\)
• D. \(\frac{\pi}{64}\)

Hint:

A is a rotation matrix, so \(A^n\) is the rotation through \(n\alpha\).

Answer 1

The Correct Option is D

By induction \(A^n = \begin{pmatrix}\cos n\alpha & -\sin n\alpha\\ \sin n\alpha & \cos n\alpha\end{pmatrix}\).
So \(A^{32} = \begin{pmatrix}\cos 32\alpha & -\sin 32\alpha\\ \sin 32\alpha & \cos 32\alpha\end{pmatrix} = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\), giving \(\cos 32\alpha = 0\), \(\sin 32\alpha = 1\).
\(32\alpha = \frac{\pi}{2} + 2k\pi \Rightarrow \alpha = \frac{\pi}{64} + \frac{k\pi}{16}\); for \(k = 0\), \(\alpha = \frac{\pi}{64}\).
Answer: (4) \(\frac{\pi}{64}\)