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  4. Let E (θ) = [cos θ&sin θ -sin θ &cos θ ] then E(α) E(β) is equal to

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Q. Let $ E\, \left(\theta\right) = \begin{bmatrix}cos\,\theta&sin\,\theta \\ -sin\,\theta &cos\,\theta \end{bmatrix} $ then $ E\left(\alpha\right)\,E\left(\beta\right) $ is equal to

J & K CETJ & K CET 2019

Solution:

Given, $E \left(\theta\right)=\left[\begin{matrix}cos\,\theta&sin\,\theta\\ -sin\,\theta&cos\,\theta\end{matrix}\right]$
Now, $E\left(\alpha\right)E\left(\beta\right)=\left[\begin{matrix}cos\,\alpha&sin\,\alpha\\ -sin\,\alpha&cos\,\alpha\end{matrix}\right] \left[\begin{matrix}cos\,\beta&sin\,\beta\\ -sin\,\beta&cos\,\beta\end{matrix}\right]$
$=\left[\begin{matrix}cos\,\alpha\,cos\,\beta-sin\,\alpha\,sin\,\beta&cos\,\alpha\,sin\,\beta+sin\,\alpha\,cos\,\beta\\ -\left(sin \,\alpha cos\,\beta+cos\,\alpha\,sin\,\beta\right)&-sin\,\alpha\,sin\,\beta+cos\,\alpha\,cos\,\beta\quad\end{matrix}\right]$
$=\left[\begin{matrix}cos\left(\alpha+\beta\right)&sin\left(\alpha+\beta\right)\\ -sin\left(\alpha+\beta\right)&cos\left(\alpha+\beta\right)\end{matrix}\right]=E\left(\alpha+\beta\right)$