For the symmetric matrix, `A=[(2,x,4),(5,3,8),(4,y,9)]` find the values of 'x' and 'y'

Answer

To find the values of 'x' and 'y' in the symmetric matrix \( A = \begin{pmatrix} 2 & x & 4 \\ 5 & 3 & 8 \\ 4 & y & 9 \end{pmatrix} \), we will use the property of symmetric matrices, which states that a matrix is symmetric if it is equal to its transpose.

### Step-by-Step Solution:

1. **Write down the matrix A and its transpose:**
\[ A = \begin{pmatrix} 2 & x & 4 \\ 5 & 3 & 8 \\ 4 & y & 9 \end{pmatrix} \]
The transpose of matrix \( A \), denoted as \( A^T \), is obtained by swapping rows with columns:
\[ A^T = \begin{pmatrix} 2 & 5 & 4 \\ x & 3 & y \\ 4 & 8 & 9 \end{pmatrix} \]

2. **Set the matrix A equal to its transpose:**
Since \( A \) is symmetric, we have:
\[ A = A^T \]
This gives us the following equations by comparing corresponding elements:
\[ \begin{pmatrix} 2 & x & 4 \\ 5 & 3 & 8 \\ 4 & y & 9 \end{pmatrix} = \begin{pmatrix} 2 & 5 & 4 \\ x & 3 & y \\ 4 & 8 & 9 \end{pmatrix} \]

3. **Compare corresponding elements:**
From the matrices, we can derive the following equations:
- From the (1,2) position: \( x = 5 \)
- From the (3,2) position: \( y = 8 \)

4. **Conclusion:**
Therefore, the values of \( x \) and \( y \) are:
\[ x = 5, \quad y = 8 \]

### Final Answer:
\[ x = 5, \quad y = 8 \]