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 1: Write down the matrices
We have:
\[ A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \]
We need to compute \( A^2 = A \times A \).

### Step 2: Set up the multiplication
To multiply two matrices, we use the formula:
\[ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]
where \( C \) is the resulting matrix, and \( n \) is the number of columns in \( A \) (or rows in \( B \)). Here, both matrices are \( 2 \times 2 \).

### Step 3: Calculate the elements of \( A^2 \)

1. **Element at (1,1)**:
\[ C_{11} = (3 \times 3) + (1 \times -1) = 9 - 1 = 8 \]

2. **Element at (1,2)**:
\[ C_{12} = (3 \times 1) + (1 \times 2) = 3 + 2 = 5 \]

3. **Element at (2,1)**:
\[ C_{21} = (-1 \times 3) + (2 \times -1) = -3 - 2 = -5 \]

4. **Element at (2,2)**:
\[ C_{22} = (-1 \times 1) + (2 \times 2) = -1 + 4 = 3 \]

### Step 4: Write the resulting matrix
Combining all the calculated elements, we have:
\[ A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \]

### Final Answer:
Thus, \( A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \).

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