Let `A+B=[{:(2,3),(5,-1):}]`, where `A` is a symmetric matrix and `B` is a skew symmetric materix, then `AxxB` is equal to

Options

A. `[{:(4,2),(1,4):}]`
B. `[{:(-4,2),(1,4):}]`
C. `[{:(4,-2),(-1,-4):}]`
D. `[{:(-4,2),(1,-4):}]`

Answer

To solve the problem, we need to find the product of two matrices \( A \) and \( B \), where \( A \) is a symmetric matrix and \( B \) is a skew-symmetric matrix, given that \( A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} \).

### Step-by-step Solution:

1. **Define the given matrix**:
Let \( P = A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} \).

2. **Find matrix \( A \)**:
Since \( A \) is symmetric, we can express it as:
\[ A = \frac{1}{2}(P + P^T) \]
First, we need to compute the transpose of \( P \):
\[ P^T = \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \]
Now, we can calculate \( A \):
\[ A = \frac{1}{2} \left( \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} + \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \right) = \frac{1}{2} \begin{pmatrix} 4 & 8 \\ 8 & -2 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 4 & -1 \end{pmatrix} \]

3. **Find matrix \( B \)**:
Since \( B \) is skew-symmetric, we can express it as:
\[ B = \frac{1}{2}(P - P^T) \]
Now, we can calculate \( B \):
\[ B = \frac{1}{2} \left( \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} - \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \right) = \frac{1}{2} \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

4. **Multiply matrices \( A \) and \( B \)**:
Now, we need to compute \( A \times B \):
\[ A \times B = \begin{pmatrix} 2 & 4 \\ 4 & -1 \end{pmatrix} \times \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]
Performing the multiplication:
- First row, first column: \( 2 \times 0 + 4 \times 1 = 4 \)
- First row, second column: \( 2 \times -1 + 4 \times 0 = -2 \)
- Second row, first column: \( 4 \times 0 + (-1) \times 1 = -1 \)
- Second row, second column: \( 4 \times -1 + (-1) \times 0 = -4 \)

Thus, we have:
\[ A \times B = \begin{pmatrix} 4 & -2 \\ -1 & -4 \end{pmatrix} \]

5. **Final Result**:
Therefore, the product \( A \times B \) is:
\[ \begin{pmatrix} 4 & -2 \\ -1 & -4 \end{pmatrix} \]