If the matrix, `A= ((2,x+2),(2x-3,x+1))` is symmetric, find the value of x

Options

A. 4
B. `-4`
C. 0
D. 5

Answer

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

\[ A = \begin{pmatrix} 2 & x + 2 \\ 2x - 3 & x + 1 \end{pmatrix} \]

is symmetric, we follow these steps:

### Step 1: Understand the condition for symmetry
A matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) is the transpose of \( A \).

### Step 2: Calculate the transpose of matrix \( A \)
The transpose of matrix \( A \) is obtained by swapping its rows and columns:

\[ A^T = \begin{pmatrix} 2 & 2x - 3 \\ x + 2 & x + 1 \end{pmatrix} \]

### Step 3: Set the matrix equal to its transpose
For the matrix to be symmetric, we need:

\[ A = A^T \]

This gives us the equation:

\[ \begin{pmatrix} 2 & x + 2 \\ 2x - 3 & x + 1 \end{pmatrix} = \begin{pmatrix} 2 & 2x - 3 \\ x + 2 & x + 1 \end{pmatrix} \]

### Step 4: Equate corresponding elements
From the equality of the matrices, we can equate the corresponding elements:

1. From the first row, first column: \( 2 = 2 \) (this is always true)
2. From the first row, second column: \( x + 2 = 2x - 3 \)
3. From the second row, first column: \( 2x - 3 = x + 2 \)
4. From the second row, second column: \( x + 1 = x + 1 \) (this is also always true)

### Step 5: Solve the equations
We can solve either of the equations from steps 2 or 3 since they are equivalent.

**Using the equation \( x + 2 = 2x - 3 \):**

\[ x + 2 = 2x - 3 \]

Rearranging gives:

\[ 2 + 3 = 2x - x \]

\[ 5 = x \]

**Using the equation \( 2x - 3 = x + 2 \):**

\[ 2x - 3 = x + 2 \]

Rearranging gives:

\[ 2x - x = 2 + 3 \]

\[ x = 5 \]

### Final Answer
Thus, the value of \( x \) for which the matrix \( A \) is symmetric is

\[ \boxed{5} \]