If `A=[(-2,4),(-1,2)]` then `A^(2)` is equal to

Options

A. null matrix
B. unit matrix
C. `[(1,0),(0,1)]`
D. `[(0,0),(0,1)]`

Explanation

To find \( A^2 \) for the matrix \( A = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix} \), we will perform matrix multiplication of \( A \) with itself.

### Step-by-Step Solution:

1. **Write down the matrix \( A \)**:
\[ A = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix} \]

2. **Set up the multiplication for \( A^2 \)**:
\[ A^2 = A \times A = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix} \times \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix} \]

3. **Calculate the elements of the resulting matrix**:
- **Element at position (1,1)**:
\[ A_{11} = (-2) \times (-2) + (4) \times (-1) = 4 - 4 = 0 \]
- **Element at position (1,2)**:
\[ A_{12} = (-2) \times (4) + (4) \times (2) = -8 + 8 = 0 \]
- **Element at position (2,1)**:
\[ A_{21} = (-1) \times (-2) + (2) \times (-1) = 2 - 2 = 0 \]
- **Element at position (2,2)**:
\[ A_{22} = (-1) \times (4) + (2) \times (2) = -4 + 4 = 0 \]

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

### Final Result:
\[ A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]