If `A=[{:(1, 3/5, x),(y, -5, -7),(-4,-7,0):}]` is a symmetric matrix then the value of x and y are ______

Options

A. `3/4` and 4
B. `5/3` and `-4`
C. `3/5` and `-4`
D. `-4` and `3/5`

Answer

To find the values of \( x \) and \( y \) in the symmetric matrix \( A = \begin{pmatrix} 1 & \frac{3}{5} & x \\ y & -5 & -7 \\ -4 & -7 & 0 \end{pmatrix} \), we will use the property of symmetric matrices, which states that \( a_{ij} = a_{ji} \) for all \( i \neq j \).

### Step-by-Step Solution:

1. **Identify the Elements of the Matrix**:
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} 1 & \frac{3}{5} & x \\ y & -5 & -7 \\ -4 & -7 & 0 \end{pmatrix} \]

2. **Use the Symmetric Matrix Property**:
For \( A \) to be symmetric, the following conditions must hold:
- \( a_{12} = a_{21} \)
- \( a_{13} = a_{31} \)
- \( a_{23} = a_{32} \)

3. **Set Up the Equations**:
- From \( a_{12} = a_{21} \):
\[ \frac{3}{5} = y \quad \text{(1)} \]
- From \( a_{13} = a_{31} \):
\[ x = -4 \quad \text{(2)} \]
- From \( a_{23} = a_{32} \):
\[ -7 = -7 \quad \text{(This is always true and does not provide new information.)} \]

4. **Solve the Equations**:
- From equation (1), we find:
\[ y = \frac{3}{5} \]
- From equation (2), we find:
\[ x = -4 \]

5. **Final Values**:
Thus, the values of \( x \) and \( y \) are:
\[ x = -4, \quad y = \frac{3}{5} \]

### Conclusion:
The values of \( x \) and \( y \) are:
\[ x = -4, \quad y = \frac{3}{5} \]