If the system of equations

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

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