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  4. If A = [ cosθ&- sinθ sinθ& cosθ], then the matrix A-50 when θ = (π/12) , is equal to :

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Q. If $A = \begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$, then the matrix $A^{-50}$ when $\theta = \frac{\pi}{12}$ , is equal to :

JEE MainJEE Main 2019Determinants

Solution:

Here, $AA^T = I$
$\Rightarrow A^{-1} =A^{T} = \begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix} $
$ A^{-n} = \begin{bmatrix}\cos\left(n\theta\right)&\sin\left(n\theta\right)\\ -\sin\left(n\theta\right)&\cos\left(n\theta\right)\end{bmatrix} $
$ \therefore A^{-50} = \begin{bmatrix}\cos\left(50\right)\theta&\sin\left(50\right)\theta\\ -\sin\left(50\right)\theta&\cos\left(50\right)\theta\end{bmatrix} $
$ = \begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\ - \frac{1}{2}& \frac{\sqrt{3}}{2}\end{bmatrix}$