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Mathematics
If A=[ beginmatrix cos α sin α - sin α cos α endmatrix ], then A2 is equal to
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Q. If $ A=\left[ \begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \\ \end{matrix} \right], $ then $ {{A}^{2}} $ is equal to
Rajasthan PET
Rajasthan PET 2001
A
$ \left[ \begin{matrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \\ \end{matrix} \right] $
B
$ \left[ \begin{matrix} {{\cos }^{2}}\alpha & -{{\sin }^{2}}\alpha \\ -{{\sin }^{2}}\alpha & {{\cos }^{2}}\alpha \\ \end{matrix} \right] $
C
$ \left[ \begin{matrix} {{\sin }^{2}}\alpha & {{\cos }^{2}}\alpha \\ -{{\cos }^{2}}\alpha & {{\sin }^{2}}\alpha \\ \end{matrix} \right] $
D
$ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
Solution:
$ A=\left[ \begin{matrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \\ \end{matrix} \right] $
$ {{A}^{2}}=A.A $
$ =\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 & \cos \alpha \sin \alpha +\sin \alpha \cos \alpha \\ -\sin \alpha \cos \alpha -\sin \alpha \cos \alpha & -{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha \\ \end{matrix} \right] $ $ =\left[ \begin{matrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \\ \end{matrix} \right] $