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Question

If $$A=\begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta\end{bmatrix}$$. Find the value $$A^2$$.


Solution

$$A^2=A\times A=\begin{bmatrix}cos2\theta & sin2\theta\\-sin2\theta & cos2\theta\end{bmatrix}\begin{bmatrix}cos2\theta & sin2\theta\\-sin2\theta & cos2\theta\end{bmatrix}$$

$$=\begin{bmatrix}cos2\theta\times cos2\theta  -sin2\theta\times sin2\theta & cos2\theta\times sin2\theta + sin2\theta\times cos2\theta \\-sin2\theta\times cos2\theta+cos2\theta\times-sin2\theta & -sin2\theta\times sin2\theta+cos2\theta\times cos2\theta\end{bmatrix}$$

$$=\begin{bmatrix}cos^2 2\theta-sin^22\theta & 2sin2\theta cos2\theta\\-2sin2\theta cos2\theta & cos^2 2\theta-sin^22\theta \end{bmatrix}$$

$$=\begin{bmatrix}cos4\theta & sin4\theta\\-sin4\theta & cos4\theta\end{bmatrix}$$

Mathematics

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