If the system of equations \(2x + ky + z = 0\), \(kx + 2y + z = 0\), \(x + y + z = 0\) has a non-zero solution, then the possible values of k are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0 or 2
A homogeneous system has a non-zero solution only when its coefficient determinant is zero.
\(\displaystyle \begin{vmatrix}2 & k & 1\\ k & 2 & 1\\ 1 & 1 & 1\end{vmatrix} = 0\)
Expanding along the first row:
\(2(2 - 1) - k(k - 1) + 1(k - 2) = 0\)
\(2 - k^{2} + k + k - 2 = 0\)
\(2k - k^{2} = 0\)
\(k(2 - k) = 0\)
\(k = 0\) or \(k = 2\)
Check: for k = 2 the first two equations become the same, 2x + 2y + z = 0.