The system of equations

$\mathrm{kx}+\mathrm{y}+\mathrm{z}=1$

$x+k y+z=k$

$x+y+z k=k^{2}$

$\Delta=\left|\begin{array}{ccc}\mathrm{K} & 1 & 1 \\ 1 & \mathrm{~K} & 1 \\ 1 & 1 & \mathrm{~K}\end{array}\right|=\mathrm{K}\left(\mathrm{K}^{2}-1\right)-1(\mathrm{~K}-1)+1(1-\mathrm{K})$

$=K^{3}-K-K+1+1-K$

$=K^{3}-3 K+2$

$=(K-1)^{2}(K+2)$

For $\mathrm{K}=1$

$\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$

But for $\mathrm{K}=-2$, at least one out of $\Delta_{1}, \Delta_{2}, \Delta_{3}$

are not zero

Hence for no soln, $K=-2$