If `A=[{:(0,1),(1,0):}]` then `A^(2)` is equal to

Options

A. `[{:(0,1),(1,0):}]`
B. `[{:(1,0),(1,0):}]`
C. `[{:(0,1),(0,1):}]`
D. `[{:(1,0),(0,1):}]`

Explanation

To find \( A^2 \) where \( A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), we will multiply the matrix \( A \) by itself.

### Step-by-Step Solution:

1. **Write down the matrix \( A \)**:
\[ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

2. **Set up the multiplication for \( A^2 \)**:
\[ A^2 = A \times A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

3. **Perform the multiplication**:
- To find the element in the first row and first column:
\[ (0 \times 0) + (1 \times 1) = 0 + 1 = 1 \]
- To find the element in the first row and second column:
\[ (0 \times 1) + (1 \times 0) = 0 + 0 = 0 \]
- To find the element in the second row and first column:
\[ (1 \times 0) + (0 \times 1) = 0 + 0 = 0 \]
- To find the element in the second row and second column:
\[ (1 \times 1) + (0 \times 0) = 1 + 0 = 1 \]

4. **Combine the results**:
\[ A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

5. **Final Result**:
\[ A^2 = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
where \( I \) is the identity matrix.