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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES3 Marks
Question
If $\text{A}=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ find x satisfying $0<\text{x}<\frac{\pi}{2}$ when $A + A^T = I$
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Answer

Given,
$\text{A}=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}=\begin{bmatrix}\cos\text{x}&-\sin\text{x}\\\sin\text{x}&\cos\text{x}\end{bmatrix}$
$\text{A}+\text{A}^\text{T}=\text{I}$
$\Rightarrow\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}+\begin{bmatrix}\cos\text{x}&-\sin\text{x}\\\sin\text{x}&\cos\text{x}\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}\cos\text{x}+\cos\text{x}&\sin\text{x}-\sin\text{x}\\-\sin\text{x}+\sin\text{x}&\cos\text{x}+\cos\text{x}\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2\cos\text{x}&0\\0&2\cos\text{x}\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$
Since, corresponding entries of equal matrices are equal, so
$2\cos\text{x}=1$
$\cos\text{x}=\frac{1}{2}$ since $0<\text{x}<\frac{\pi}{2}$
So,
$\text{x}=\frac{\pi}{3}.$

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