Find \(A^2\)

Solution:

Step 1: Multiply \(A \cdot A\)

\[ A^2 = \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \]

Step 2: Multiply

\[ = \begin{bmatrix} \cos^2 2\theta – \sin^2 2\theta & \cos 2\theta \sin 2\theta + \sin 2\theta \cos 2\theta \\ -\sin 2\theta \cos 2\theta – \cos 2\theta \sin 2\theta & -\sin^2 2\theta + \cos^2 2\theta \end{bmatrix} \]

Step 3: Use identities

\[ \cos^2 x – \sin^2 x = \cos 2x \] \[ 2\sin x \cos x = \sin 2x \] So, \[ A^2 = \begin{bmatrix} \cos 4\theta & \sin 4\theta \\ -\sin 4\theta & \cos 4\theta \end{bmatrix} \]

Final Answer:

\[ \boxed{ \begin{bmatrix} \cos 4\theta & \sin 4\theta \\ -\sin 4\theta & \cos 4\theta \end{bmatrix} } \]