Question

If \( A = \begin{bmatrix} 10 & 2k + 5 \\ 3k - 3 & k + 5 \end{bmatrix} \) is a symmetric matrix, the value of \( k \) is ________________.

• A. 8
• B. 5
• C. -0.4
• D. \( \frac{1 + \sqrt{1561}}{12} \)

Hint:

For a matrix to be symmetric, the off-diagonal elements must be equal, meaning \( A_{1,2} = A_{2,1} \).

Answer 1

The Correct Option is A

For a matrix to be symmetric, the elements on the off-diagonal must be equal. That is, the element in the \( (1,2) \)-position must be equal to the element in the \( (2,1) \)-position. In this case, the matrix \( A \) is given as: \[ A = \begin{bmatrix} 10 & 2k + 5 \\ \end{bmatrix} \] For \( A \) to be symmetric, the element \( 2k + 5 \) (position \( (1,2) \)) must be equal to \( 3k - 3 \) (position \( (2,1) \)). Therefore, we set up the equation: \[ 2k + 5 = 3k - 3 \] Now, solve for \( k \): \[ 2k + 5 = 3k - 3
5 + 3 = 3k - 2k
8 = k \] Thus, the value of \( k \) is 8. Final Answer:
\[ \boxed{8} \]