If A = `[(2,-1),(-1,2)]` and I is the unit matrix of order 2 then `A^(2) ` equals

Options

A. 4A-3l
B. 3A-4l
C. A-l
D. A+l

Answer

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

### Step-by-Step Solution:

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

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

3. **Perform the multiplication:**
- To find the element in the first row, first column:
\[ (2 \times 2) + (-1 \times -1) = 4 + 1 = 5 \]
- To find the element in the first row, second column:
\[ (2 \times -1) + (-1 \times 2) = -2 - 2 = -4 \]
- To find the element in the second row, first column:
\[ (-1 \times 2) + (2 \times -1) = -2 - 2 = -4 \]
- To find the element in the second row, second column:
\[ (-1 \times -1) + (2 \times 2) = 1 + 4 = 5 \]

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

### Final Result:
\[ A^2 = \begin{pmatrix} 5 & -4 \\ -4 & 5 \end{pmatrix} \]