If A = [ cos α − sin α sin α cos α ] , then show that $ A 2 = [ cos 2 α − sin 2 α sin 2 α cos 2 α ] $

Question

If A = [ cos α − sin α ​ sin α cos α ​ ] , then show that $ A 2 = [ cos 2 ​ α − sin 2 ​ α ​ sin 2 ​ α cos 2 ​ α ​ ] $,

MARKS App · Accepted Answer

\begin{aligned} \mathrm{A}^2 &=\left[\begin{array}{cc} \cos \alpha & \sin \alpha \ -\sin \alpha & \cos \alpha \end{array}ight]\left[\begin{array}{cc} \cos \alpha & \sin \alpha \ -\sin \alpha & \cos \alpha \end{array}ight] \ &=\left[\begin{array}{cc} \cos ^2 \alpha-\sin ^2 \alpha & \sin \alpha \cos \alpha+\sin \alpha \cos \alpha \ -\sin \alpha \cos \alpha-\sin \alpha \cos \alpha & -\sin ^2 \alpha+\cos ^2 \alpha \end{array}ight] \ &=\left[\begin{array}{cc} \cos 2 \alpha & \sin 2 \alpha \ -\sin 2 \alpha & \cos 2 \alpha \end{array}ight] \end{aligned} $$

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