if-a-4x-22x-3x-1-is-a-symmetric-matrix-then-x

To determine the value of \( x \) that makes the matrix \( A = \begin{bmatrix} 4 & x \\ 2x - 3 & x - 1 \end{bmatrix} \) symmetric, we need to recall that a matrix is symmetric if it is equal to its transpose. 

For a matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), it is symmetric if \( a = a \), \( b = c \), and \( d = d \). 

### Step 1: Set up the conditions for symmetry
Given \( A = \begin{bmatrix} 4 & x \\ 2x - 3 & x - 1 \end{bmatrix} \), for \( A \) to be symmetric:

1. The element in the top right (\( x \)) must equal the element in the bottom left (\( 2x - 3 \)).

\[
x = 2x - 3
\]

### Step 2: Solve the equation
Now, solve for \( x \):

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

### Final Answer
The value of \( x \) that makes the matrix symmetric is \( x = 3 \).

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the determinant of the matrix when \( x = 3 \)?
2. Can you find the eigenvalues of the symmetric matrix with \( x = 3 \)?
3. How would you find the inverse of the symmetric matrix when \( x = 3 \)?
4. What is the trace of the matrix when \( x = 3 \)?
5. Can you verify the matrix is positive definite for \( x = 3 \)?

**Tip:** Symmetric matrices have some nice properties, such as real eigenvalues and orthogonal eigenvectors, making them particularly useful in various applications.

Explore how to determine the value of x that makes the matrix A = [4 x; 2x - 3 x - 1] symmetric. Detailed steps and solution provided. Learn about symmetric matrices and their properties. Suitable for advanced high school students.

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