If `A=[(3,1),(-1,2)]` then `A^(2)=`

Options

A. `[(8,-5),(-5,3)]`
B. `[(8,-5),(5,3)]`
C. `[(8,-5),(-5,-3)]`
D. `[(8,5),(-5,3)]`

Answer

To find \( A^2 \) where \( A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \), we will perform matrix multiplication of \( A \) with itself.

### Step-by-step Solution:

1. **Write down the matrix \( A \)**:
\[ A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \]

2. **Set up the multiplication**:
We need to calculate \( A^2 = A \times A \):
\[ A^2 = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \times \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \]

3. **Calculate the elements of the resulting matrix**:
- **First row, first column**:
\[ (3 \times 3) + (1 \times -1) = 9 - 1 = 8 \]
- **First row, second column**:
\[ (3 \times 1) + (1 \times 2) = 3 + 2 = 5 \]
- **Second row, first column**:
\[ (-1 \times 3) + (2 \times -1) = -3 - 2 = -5 \]
- **Second row, second column**:
\[ (-1 \times 1) + (2 \times 2) = -1 + 4 = 3 \]

4. **Combine the results into the final matrix**:
\[ A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \]

### Final Answer:
\[ A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \]