A) \[\left( \begin{matrix} {{\cos }^{10}}\alpha & {{\sin }^{10}}\alpha \\ -{{\sin }^{10}}\alpha & {{\cos }^{10}}\alpha \\ \end{matrix} \right)\]
B) \[\left( \begin{matrix} {{\cos }^{10}}\alpha & -{{\sin }^{10}}\alpha \\ {{\sin }^{10}}\alpha & {{\cos }^{10}}\alpha \\ \end{matrix} \right)\]
C) \[\left( \begin{matrix} {{\cos }^{10}}\alpha & {{\sin }^{10}}\alpha \\ -{{\sin }^{10}}\alpha & -{{\cos }^{10}}\alpha \\ \end{matrix} \right)\]
D) \[\left( \begin{matrix} \cos 10\alpha & \sin 10\alpha \\ -\sin 10\alpha & \cos 10\alpha \\ \end{matrix} \right)\]
E) \[\left( \begin{matrix} \cos 10\alpha & -\sin 10\alpha \\ -\sin 10\alpha & -\cos 10\alpha \\ \end{matrix} \right)\]
Correct Answer: D
Solution :
\[A=\left( \begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \\ \end{matrix} \right),{{A}^{10}}=?\] \[{{A}^{2}}=\left( \begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \\ \end{matrix} \right)\left( \begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \\ \end{matrix} \right)\] \[=\left( \begin{matrix} {{\cos }^{2}}\alpha -{{\sin }^{2}}\alpha & 2\sin \alpha \cos \alpha \\ -2\sin \alpha .\cos \alpha & {{\cos }^{2}}\alpha -{{\sin }^{2}}\alpha \\ \end{matrix} \right)\] \[=\left( \begin{matrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \\ \end{matrix} \right)\] Similarly, \[{{A}^{10}}=\left( \begin{matrix} \cos 10\alpha & \sin 10\alpha \\ -\sin 10\alpha & \cos 10\alpha \\ \end{matrix} \right)\]
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