Question:
If the system of equations $x-k y-z=0, k x-y-z=0, x+y-z=0$ has a non-zero solution then the values of $k$ are________
Solution:
The system of homogeneous equations $x-k y-z=0, k x-y-z=0$ and $x+y-z=0$ has a non-zero solution or an infinite many solutions.
$\therefore \Delta=\left|\begin{array}{ccc}1 & -k & -1 \\ k & -1 & -1 \\ 1 & 1 & -1\end{array}\right|=0$
$\Rightarrow 1(1+1)+k(-k+1)-1(k+1)=0$
$\Rightarrow 2-k^{2}+k-k-1=0$
$\Rightarrow k^{2}=1$
$\Rightarrow k=\pm 1$
Thus, the values of $k$ are $-1$ and 1 .
If the system of equations $x-k y-z=0, k x-y-z=0, x+y-z=0$ has a non-zero solution then the values of $k$ are $-1$ and 1