If in matrix `[{:(3,2x,-5),(-4,0,y),(-5,3,7):}]` is symmetric, then :

Options

A. `x=2, y=-3`
B. `x=2 y=3`
C. `x=-2, y=-3`
D. `x=-2y,y=3`

Answer

To determine the values of \( x \) and \( y \) in the given matrix

\[ A = \begin{pmatrix} 3 & 2x & -5 \\ -4 & 0 & y \\ -5 & 3 & 7 \end{pmatrix} \]

we need to use the property of symmetric matrices, which states that a matrix is symmetric if it is equal to its transpose, i.e., \( A = A^T \).

### Step 1: Write the Transpose of the Matrix

The transpose of matrix \( A \) is obtained by swapping rows with columns:

\[ A^T = \begin{pmatrix} 3 & -4 & -5 \\ 2x & 0 & 3 \\ -5 & y & 7 \end{pmatrix} \]

### Step 2: Set the Matrix Equal to Its Transpose

Since \( A \) is symmetric, we have:

\[ \begin{pmatrix} 3 & 2x & -5 \\ -4 & 0 & y \\ -5 & 3 & 7 \end{pmatrix} = \begin{pmatrix} 3 & -4 & -5 \\ 2x & 0 & 3 \\ -5 & y & 7 \end{pmatrix} \]

### Step 3: Equate Corresponding Elements

From the equality of the two matrices, we can equate corresponding elements:

1. From the first row, second column:
\[ 2x = -4 \]

2. From the second row, third column:
\[ y = 3 \]

### Step 4: Solve for \( x \)

Now, we solve the equation \( 2x = -4 \):

\[ x = \frac{-4}{2} = -2 \]

### Step 5: State the Values of \( x \) and \( y \)

Thus, we find:

\[ x = -2 \quad \text{and} \quad y = 3 \]

### Final Answer

The values of \( x \) and \( y \) are:

\[ x = -2, \quad y = 3 \]

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