Question

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 value of $k$

Answer 1

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

$\left|\begin{array}{ccc}1& -k& -1\\ k& -1& -1\\ 1& 1& -1\end{array}\right|=0$

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

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

$\Rightarrow k^2=1$

$\therefore k=\pm 1$