Question

If \( \begin{bmatrix} 2k+7 & 2k^2 - 5k + 3 \\ 3k - 3 & 15 \end{bmatrix} \) is a \(2 \times 2\) symmetric matrix, then the value of \(k\) is .........

• A. 4
• B. -1
• C. 3
• D. 0

Hint:

For symmetric matrices, equate the off-diagonal elements \((i,j)\) and \((j,i)\) to form an equation and solve for the variable.

Answer 1

The Correct Option is C

A symmetric matrix satisfies the condition that the element at position \((i,j)\) is equal to the element at position \((j,i)\).
So, for the matrix to be symmetric: \[ 2k^2 - 5k + 3 = 3k - 3 \] Rearranging: \[ 2k^2 - 5k + 3 - 3k + 3 = 0 \] \[ 2k^2 - 8k + 6 = 0 \] Divide the entire equation by 2: \[ k^2 - 4k + 3 = 0 \] Solving the quadratic: \[ k = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} \] \[ ⇒ k = 3 \text{ or } k = 1 \] Now check both values to see which make the matrix symmetric:
Try \(k = 3\):
- Top right = \(2(3)^2 - 5(3) + 3 = 18 - 15 + 3 = 6\)
- Bottom left = \(3(3) - 3 = 6\)
They match \(⇒ \) Matrix is symmetric.
So, the correct value is \( \boxed{3} \).