If A = `[{:(2, x - 2),(2x +3, x -3):}]` is a symmetric matrix, then the value of x is

Options

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

Answer

To determine the value of \( x \) for which the matrix

\[ A = \begin{pmatrix} 2 & x - 2 \\ 2x + 3 & x - 3 \end{pmatrix} \]

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

### Step-by-step Solution:

1. **Write down the matrix \( A \)**:
\[ A = \begin{pmatrix} 2 & x - 2 \\ 2x + 3 & x - 3 \end{pmatrix} \]

2. **Find the transpose of matrix \( A \)**:
The transpose of a matrix is obtained by swapping its rows with columns. Therefore,
\[ A^T = \begin{pmatrix} 2 & 2x + 3 \\ x - 2 & x - 3 \end{pmatrix} \]

3. **Set the matrix \( A \) equal to its transpose \( A^T \)**:
\[ \begin{pmatrix} 2 & x - 2 \\ 2x + 3 & x - 3 \end{pmatrix} = \begin{pmatrix} 2 & 2x + 3 \\ x - 2 & x - 3 \end{pmatrix} \]

4. **Equate corresponding elements**:
From the equality of the matrices, we get the following equations:
- From the first row, first column: \( 2 = 2 \) (which is always true).
- From the first row, second column: \( x - 2 = 2x + 3 \)
- From the second row, first column: \( 2x + 3 = x - 2 \)
- From the second row, second column: \( x - 3 = x - 3 \) (which is also always true).

5. **Solve the first equation**:
\[ x - 2 = 2x + 3 \]
Rearranging gives:
\[ x - 2x = 3 + 2 \implies -x = 5 \implies x = -5 \]

6. **Verify with the second equation**:
\[ 2x + 3 = x - 2 \]
Substitute \( x = -5 \):
\[ 2(-5) + 3 = -5 - 2 \implies -10 + 3 = -5 - 2 \implies -7 = -7 \text{ (True)} \]

7. **Conclusion**:
The value of \( x \) for which the matrix \( A \) is symmetric is:
\[ \boxed{-5} \]