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ISI2015-MMA-63

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Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ – \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ is

  1. $\begin{pmatrix} \cos \theta & \sin \theta \\ – \sin \theta & \cos \theta \end{pmatrix}$
  2. $\begin{pmatrix} 1& 0 \\ 0 & 1 \end{pmatrix}$
  3. $\begin{pmatrix} \cos^{30} \theta & \sin^{30} \theta \\ – \sin^{30} \theta & \cos^{30} \theta \end{pmatrix}$
  4. $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
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Answer: $B$

This is an example of a rotation matrix which takes any vector and then rotates it by $\frac{1}{67^{th}}$  of a revolution.

$2010 = 67 * 30 $

Therefore, the answer should be an identity matrix.

Hence, option $B$ is the right answer, since its the given identity matrix.
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