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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES3 Marks
Question
If $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix},$ and $\text{A}^{-1}=\text{A}',$ find value of $\alpha.$
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Answer

We have, $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}$ and $\text{A}'=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$ Also, $\text{A}^{-1}=\text{A}'$ $\Rightarrow\ \text{AA}^{-1}=\text{AA}'$$\Rightarrow\ \text{I}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$
$\Rightarrow\ \begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}\cos^2\alpha+\sin^2\alpha&0\\0&\sin^2\alpha+\cos^2\alpha\end{bmatrix}$
By Using equality of matrices, we get $\cos^2\alpha+\sin^2\alpha=1$ Which is true for all real values of $\alpha.$

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