If `A(alpha)=[(cos alpha, sin alpha),(-sin alpha, cos alpha)]` then the matrix `A^(2)(alpha)` is

Options

A. `A(2 alpha)`
B. `A(alpha)`
C. `A(3 alpha)`
D. `A(4 alpha)`

Explanation

To find the matrix \( A^2(\alpha) \) where

\[ A(\alpha) = \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix}, \]

we will multiply the matrix \( A(\alpha) \) by itself.

### Step-by-Step Solution:

1. **Write down the matrix multiplication**:
We need to compute \( A(\alpha) \times A(\alpha) \):
\[ A^2(\alpha) = A(\alpha) \cdot A(\alpha) = \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix} \cdot \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix}. \]

2. **Calculate the elements of the resulting matrix**:
- **First row, first column**:
\[ (\cos \alpha)(\cos \alpha) + (\sin \alpha)(-\sin \alpha) = \cos^2 \alpha - \sin^2 \alpha. \]
- **First row, second column**:
\[ (\cos \alpha)(\sin \alpha) + (\sin \alpha)(\cos \alpha) = \cos \alpha \sin \alpha + \sin \alpha \cos \alpha = 2 \cos \alpha \sin \alpha. \]
- **Second row, first column**:
\[ (-\sin \alpha)(\cos \alpha) + (\cos \alpha)(-\sin \alpha) = -\sin \alpha \cos \alpha - \sin \alpha \cos \alpha = -2 \sin \alpha \cos \alpha. \]
- **Second row, second column**:
\[ (-\sin \alpha)(\sin \alpha) + (\cos \alpha)(\cos \alpha) = -\sin^2 \alpha + \cos^2 \alpha = \cos^2 \alpha - \sin^2 \alpha. \]

3. **Combine the results into the final matrix**:
Putting all the calculated elements together, we have:
\[ A^2(\alpha) = \begin{pmatrix} \cos^2 \alpha - \sin^2 \alpha & 2 \cos \alpha \sin \alpha \\ -2 \sin \alpha \cos \alpha & \cos^2 \alpha - \sin^2 \alpha \end{pmatrix}. \]

4. **Simplify the matrix**:
We can recognize that:
\[ \cos^2 \alpha - \sin^2 \alpha = \cos(2\alpha) \]
and
\[ 2 \cos \alpha \sin \alpha = \sin(2\alpha). \]
Therefore, we can rewrite the matrix as:
\[ A^2(\alpha) = \begin{pmatrix} \cos(2\alpha) & \sin(2\alpha) \\ -\sin(2\alpha) & \cos(2\alpha) \end{pmatrix}. \]

### Final Answer:
\[ A^2(\alpha) = \begin{pmatrix} \cos(2\alpha) & \sin(2\alpha) \\ -\sin(2\alpha) & \cos(2\alpha) \end{pmatrix}. \]