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Question
If \[A = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}\] then AT + A = I2, if
Options
θ = n π, n ∈ Z
θ = (2n + 1) \[\frac{\pi}{2}\] n ∈ Z
θ = 2n π +\[\frac{\pi}{3}\] n ∈ Z
none of these
Solution
θ = 2nπ + \[\frac{\pi}{3}\]n ∈ Z
\[Here, \]
\[A = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix} \]
\[ \Rightarrow A^T = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\]
\[Now, \]
\[ A^T + A = I_2 \]
\[ \Rightarrow \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix} + \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2\cos \theta & 0 \\ 0 & 2\cos \theta\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow 2\cos \theta = 1\]
\[ \Rightarrow \cos \theta = \frac{1}{2}\]
\[ \Rightarrow \cos \theta = \cos\frac{\pi}{3}\]
\[ \Rightarrow \theta = 2n\pi \pm \frac{\pi}{3} \left( n \in Z \right)\]
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