The given system of equations are $\mathrm{kx}+\mathrm{y}+\mathrm{z}=\mathrm{k}-1$
$x+k y+z=k-1$
$\mathrm{x}+\mathrm{y}+\mathrm{kz}=\mathrm{k}-1$
$A=\left[\begin{array}{lll}k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k\end{array}\right], B=\left[\begin{array}{l}k-1 \\ k-1 \\ k-1\end{array}\right]$ and $\mathrm{x}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$
Now, $|A|=\left|\begin{array}{lll}k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k\end{array}\right|$
$=k\left(k^{2}-1\right)-1(k-1)+1(1-k)$
$=k^{3}-k-k+1+1-k$
$=k^{3}-3 k+2$
The given system of equations has no solution, if
$|A|=0$
$\Rightarrow k^{3}-3 k+2=0$
$\Rightarrow(k-1)^{2}(k+2)=0$
$\Rightarrow k=1$ or $k=-2$