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  4. If A = [ cos θ&- sin θ sin θ& cos θ], then A3 is equal to

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Q. If $A = \begin{bmatrix}\cos \theta&-\sin \theta\\ \sin \theta&\cos \theta\end{bmatrix}, $ then $ A^3 $ is equal to

AMUAMU 2013Matrices

Solution:

Given, $A=\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}$
$A^{2}=A \cdot A=\begin{bmatrix}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta\end{bmatrix}\begin{bmatrix}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta\end{bmatrix} $
$=\begin{bmatrix}\cos ^{2} \theta-\sin ^{2} \theta & -\sin \theta \cos \theta-\cos \theta \sin \theta \\\cos \theta \sin \theta+\sin \theta \cos \theta& -\sin ^{2} \theta+\cos ^{2} \theta\end{bmatrix} $
$=\begin{bmatrix}\cos 2 \theta & -2 \sin \theta \cos \theta \\ 2 \sin \theta \cos\theta & \cos 2 \theta\end{bmatrix} $
$=\begin{bmatrix}\cos 2 \theta & -\sin 2 \theta \\\sin 2 \theta & \cos 2 \theta\end{bmatrix} $
Similarly, $ A^{3}=\begin{bmatrix}\cos 3 \theta & -\sin 3 \theta \\\sin 3 \theta & \cos 3 \theta\end{bmatrix}$