If A = [ cases cosθ & - sinθ sinθ & cosθ cases ] , then the matrix...

A = \left[$\begin{cases}cos\theta & - sin\theta \\ sin\theta & cos\theta\end{cases}$\right] ⇒ \left|A\right| = \left|$\begin{cases}cos\theta & - sin\theta \\ sin\theta & cos\theta\end{cases}$\right| = \cos^{2}\theta + \sin^{2}\theta = 1$<br/><br/> And$adj\left(A\right) = \left[$\begin{cases}cos\theta & sin\theta \\ - sin\theta & cos\theta\end{cases}$\right]$<br/><br/>$⇒ {A}^{- 1} = \dfrac{1}{\left|A\right|}adj\left(A\right) = \left[$\begin{cases}cos\theta & sin\theta \\ - sin\theta & cos\theta\end{cases}$\right]$<br/><br/>$⇒ {A}^{- 2} = {\left({A}^{- 1}\right)}^{2} = \left({A}^{- 1}\right)\left({A}^{- 1}\right)$<br/><br/>$= \left[$\begin{cases}cos\theta & sin\theta \\ - sin\theta & cos\theta\end{cases}$\right]\left[$\begin{cases}cos\theta & sin\theta \\ - sin\theta & cos\theta\end{cases}$\right]$<br/><br/>$= \left[$\begin{cases}\cos^{2}\theta - \sin^{2}\theta & 2sin\theta cos\theta \\ - 2sin\theta cos\theta & \cos^{2}\theta - \sin 2\theta\end{cases}$\right]$<br/><br/>$= \left[$\begin{cases}\cos 2\theta & \sin 2\theta \\ - \sin 2\theta & \cos 2\theta\end{cases}$\right]$<br/><br/> It is visible that<br/><br/>${\left({A}^{- 1}\right)}^{n} = \left[$\begin{cases}cosn\theta & sinn\theta \\ - sinn\theta & cosn\theta\end{cases}$\right]$<br/><br/>$⇒ {A}^{- 50} = {\left({A}^{- 1}\right)}^{50} = \left[$\begin{cases}\cos 50\theta & \sin 50\theta \\ - \sin 50\theta & \cos 50\theta\end{cases}$\right]$<br/><br/> Now at$\theta = \dfrac{\pi}{12}, \sin 50\theta = sin\dfrac{50\pi}{12} = sin\dfrac{\pi}{6} = \dfrac{1}{2}$<br/><br/>$\cos 50\theta = cos\dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}$<br/><br/>$⇒ {A}^{- 50} = \left[$\begin{cases}\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ \dfrac{- 1}{2} & \dfrac{\sqrt{3}}{2}\end{cases}$\right]