If the system of equations: $x - ky - z = 0, kx - y - z = 0, x + y - z = 0$ has a non-zero solutions, then find the possible values of $k$.

$1$ or $-1$

The correct answer is Option (2) → $1$ or $-1$ ##

For the given homogeneous system of equations to have non-zero solution determinant of coefficient matrix should be zero.

$\begin{vmatrix} 1 & -k & -1 \\ k & -1 & -1 \\ 1 & 1 & -1 \end{vmatrix} = 0$

$\Rightarrow 1(1 + 1) + k(-k + 1) - 1(k + 1) = 0$

$\Rightarrow 2 - k^2 + k - k - 1 = 0$

$\Rightarrow k^2 = 1$

$\Rightarrow k = \pm 1$