📘 Question
If
\[
A =
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}
\]
then \(A^T + A = I_2\), if
(a) \(\theta = n\pi\)
(b) \(\theta = \frac{(2n+1)\pi}{2}\)
(c) \(\theta = 2n\pi + \frac{\pi}{3}\)
(d) none of these
✏️ Step-by-Step Solution
Step 1: Find transpose
\[
A^T =
\begin{bmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{bmatrix}
\]
Step 2: Add matrices
\[
A^T + A =
\begin{bmatrix}
2\cos\theta & 0 \\
0 & 2\cos\theta
\end{bmatrix}
\]
Step 3: Compare with identity
\[
2\cos\theta = 1
\Rightarrow \cos\theta = \frac{1}{2}
\]
Step 4: General solution
\[
\theta = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z}
\]
Step 5: Match options
Given options do not include full general solution.
✅ Final Answer
\[
\boxed{(d)\; \text{none of these}}
\]
💡 Key Concept
For rotation matrix:
\[
A^T + A = 2\cos\theta \cdot I
\]
Always reduce to a trigonometric equation.