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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES3 Marks
Question
If $\text{A}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix},$ then show that $\text{A}^2=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}.$
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Answer

We have, $\text{A}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}$ $\therefore\ \text{A}^2=\text{A}.\text{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}$
$=\begin{bmatrix}\cos^2\theta-\sin^2\theta&\cos\theta\sin\theta+\sin\theta\cos\theta\\-\sin\theta\cos\theta-\cos\theta\sin\theta&-\sin^2\theta+\cos^2\theta\end{bmatrix}$
$=\begin{bmatrix}\cos2\theta&2\sin\theta\cos\theta\\2\sin\theta\cos\theta&\cos2\theta\end{bmatrix}$
$=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}$
Hence proved.

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