Let $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ If $A^T+A=I$ , then:

The correct option is:
$\theta =2n\pi +\frac{\pi }{3},n\in \mathbb{Z}$

Step 1: Understand the Given Matrices
We are given a 2×2 matrix $A$ defined as:
$A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
And the identity matrix $I$ of order 2:
$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Step 2: Find the Transpose of Matrix A
The transpose of matrix $A$, denoted by $A^T$, is obtained by swapping its rows and columns:
$A^T=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$

Step 3: Apply the Given Equation
We are given the matrix equation:
$A^T+A=I$
Substituting the matrices into the equation, we get:
$\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Step 4: Perform Matrix Addition
Adding the corresponding elements of the matrices on the left-hand side:
$\left[\begin{array}{cc}\cos \theta +\cos \theta & \sin \theta +(-\sin \theta )\\ -\sin \theta +\sin \theta & \cos \theta +\cos \theta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Simplifying this yields:
$\left[\begin{array}{cc}2\cos \theta & 0\\ 0& 2\cos \theta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Step 5: Solve for θ
By equating the corresponding elements from both matrices, we obtain the algebraic relation:
$2\cos \theta =1$
Dividing both sides by 2:
$\cos \theta =\frac{1}{2}$
Since $\cos \frac{\pi }{3}=\frac{1}{2}$, the principal value is $\theta =\frac{\pi }{3}$.
The general solution for $\cos \theta =\cos \alpha$ is given by $\theta =2n\pi \pm \alpha$, where $n\in \mathbb{Z}$.
Considering the positive branch of the general solution, we have:
$\theta =2n\pi +\frac{\pi }{3},n\in \mathbb{Z}$