Question

The number of values of $k$ for which the following system of linear equations: \[ \begin{pmatrix} 4 & k & 2\\ k & 4 & 1\\ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \] possess a non-zero solution, is _______

• A. 1
• B. 2
• C. 3
• D. Infinite

Hint:

For any homogeneous system $A X = 0$:
Non-trivial/non-zero solution $\iff \det(A) = 0$.
Setting the determinant of the $3 \times 3$ matrix to zero results in a quadratic equation in $k$, which yields exactly 2 roots.

Answer 1

The Correct Option is B