If A = `[{:(cos theta,sin theta),(-sin theta,cos theta):}]` then what is `A^(3)` equal to ?

Options

A. `[{:(cos 3theta,sin 3theta),(-sin 3theta,cos 3theta):}]`
B. `[{:(cos^(3)theta,sin^(3)theta),(-sin^(3)theta,cos^(3)theta):}]`
C. `[{:(cos 3theta,-sin 3theta),(sin 3theta,cos 3theta):}]`
D. `[{:(cos^(3)theta,-sin^(3)theta),(sin^(3)theta,cos^(3)theta):}]`

Explanation

To find \( A^3 \) where \( A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \), we can use the property of rotation matrices. The matrix \( A \) represents a rotation by angle \( \theta \).

### Step-by-step Solution:

1. **Understanding the Matrix**:
The matrix \( A \) is a rotation matrix that rotates a vector in the plane by an angle \( \theta \). The general form of a rotation matrix is:
\[ A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

2. **Using the Property of Rotation Matrices**:
The property of rotation matrices states that:
\[ A^n = \begin{pmatrix} \cos(n\theta) & \sin(n\theta) \\ -\sin(n\theta) & \cos(n\theta) \end{pmatrix} \]
This means that raising the rotation matrix \( A \) to the power \( n \) results in a rotation by angle \( n\theta \).

3. **Calculating \( A^3 \)**:
For \( n = 3 \):
\[ A^3 = \begin{pmatrix} \cos(3\theta) & \sin(3\theta) \\ -\sin(3\theta) & \cos(3\theta) \end{pmatrix} \]

4. **Final Result**:
Thus, the result for \( A^3 \) is:
\[ A^3 = \begin{pmatrix} \cos(3\theta) & \sin(3\theta) \\ -\sin(3\theta) & \cos(3\theta) \end{pmatrix} \]