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JEE MainMathematicsModerateMatrix and determinant

If $A = \begin{bmatrix}
cos\theta & - sin\theta \\
sin\theta & cos\theta
\end{bmatrix}thenthematrixA−50when then the matrix A^-50 when \theta = \frac{\pi}{12}$ is
equal to

Show answer and solution
A = \begin{bmatrix} \cos\theta & - \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix},{adj}A = \begin{bmatrix} \cos\theta & \sin\theta \\ - \sin\theta & \cos\theta \end{bmatrix}$ $\Rightarrow A^{- 1} = \begin{bmatrix} cos\theta & sin\theta \\ - sin\theta & cos\theta \end{bmatrix}$ $\left( A^{- 1} \right)^{20} = \begin{bmatrix} cos2\theta & sin2\theta \\ - sin2\theta & cos2\theta \end{bmatrix}$ $A^{- 50} = \begin{bmatrix} cos( - 50\theta) & - sin( - 50\theta) \\ sin( - 50\theta) & cos( - 50\theta) \end{bmatrix}$ $A^{- 50}at\ \theta = \frac{\pi}{12}\ is\ \begin{bmatrix} cos\frac{25\pi}{6} & sin\frac{25\pi}{6} \\ - sin\frac{25\pi}{6} & cos\frac{25\pi}{6} \end{bmatrix}$ $= \begin{bmatrix} cos\frac{\pi}{6} & sin\frac{\pi}{6} \\ - sin\frac{\pi}{6} & cos\frac{\pi}{6} \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{- 1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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