Find AAT
Given:
\[ A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \]
Step 1: Find AT
\[ A^T = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \]
Step 2: Compute AAT
\[ AA^T = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \]
\[ AA^T = \begin{bmatrix} \cos^2 x + \sin^2 x & \cos x \sin x – \sin x \cos x \\ \sin x \cos x – \cos x \sin x & \sin^2 x + \cos^2 x \end{bmatrix} \]
Step 3: Simplify
\[ \cos^2 x + \sin^2 x = 1,\quad \cos x \sin x – \sin x \cos x = 0 \]
\[ AA^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \]
Final Answer:
\[ AA^T = I \]
Hence, A is an orthogonal matrix.