If A is a symmetric matrix and B is a skew-symmetric matrix such that
`A + B = [{:(2,3),(5,-1):}]`, then AB 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 matrices A and B given that A is symmetric, B is skew-symmetric, and their sum is a specific matrix. Let's break down the solution step by step.

### Step 1: Define the matrices A and B
Given:
- A is a symmetric matrix, which means \( A^T = A \).
- B is a skew-symmetric matrix, which means \( B^T = -B \).
- We know that \( A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} \).

### Step 2: Find the transpose of the sum
Taking the transpose of both sides of the equation \( A + B \):
\[ (A + B)^T = A^T + B^T \]
Substituting the properties of A and B:
\[ A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} \Rightarrow A + B = \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \]

### Step 3: Set up equations for A and B
We now have two equations:
1. \( A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} \) (Equation 1)
2. \( A - B = \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \) (Equation 2)

### Step 4: Add the two equations
Adding Equation 1 and Equation 2:
\[ (A + B) + (A - B) = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} + \begin{pmatrix} 2 & 5 \\ 3 & -1 \end{pmatrix} \]
This simplifies to:
\[ 2A = \begin{pmatrix} 4 & 8 \\ 8 & -2 \end{pmatrix} \]
Dividing by 2:
\[ A = \begin{pmatrix} 2 & 4 \\ 4 & -1 \end{pmatrix} \]

### Step 5: Substitute A back to find B
Now substitute A back into Equation 1 to find B:
\[ B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} - A \]
Calculating B:
\[ B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix} - \begin{pmatrix} 2 & 4 \\ 4 & -1 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

### Step 6: Calculate the product AB
Now we can find the product \( AB \):
\[ AB = \begin{pmatrix} 2 & 4 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]
Calculating the product:
- First row, first column: \( 2 \cdot 0 + 4 \cdot 1 = 4 \)
- First row, second column: \( 2 \cdot -1 + 4 \cdot 0 = -2 \)
- Second row, first column: \( 4 \cdot 0 + (-1) \cdot 1 = -1 \)
- Second row, second column: \( 4 \cdot -1 + (-1) \cdot 0 = -4 \)

Thus,
\[ AB = \begin{pmatrix} 4 & -2 \\ -1 & -4 \end{pmatrix} \]

### Final Answer
The product \( AB \) is:
\[ \begin{pmatrix} 4 & -2 \\ -1 & -4 \end{pmatrix} \]