Question

If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z = 0 has a non-zero solution then the values of k are __________________.

Answer 1

The system of homogeneous equations x − ky − z = 0, kx − y − z = 0 and x + y − z = 0 has a non-zero solution or an infinite many solutions.

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

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

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

$\Rightarrow k^2=1$

$\Rightarrow k=\pm 1$

Thus, the values of k are −1 and 1.

If the system of equations x − ky − z = 0, kx − y − z = 0, x + y − z = 0 has a non-zero solution then the values of k are __−1 and 1__.