The system of equations $\alpha x+y+z=\alpha -1$ $x+\alpha y+z=\alpha -1\text{ and }x+y+\alpha z=\alpha -1$ has no  solution, if a, is 

$\text{ Let }\mathrm{\Delta }=\left|\begin{array}{l}\alpha \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \end{array}\right|={\alpha }^3-3\alpha +2=0$

$\begin{array}{l}\Rightarrow (\alpha -1)\left({\alpha }^2+\alpha -2\right)=0\\ \Rightarrow (\alpha -1{)}^2(\alpha +2)=0\Rightarrow \alpha =1\text{ or }-2\end{array}$

$\alpha =1\Rightarrow x+y+z=0$ giving infinite number of solutions  and $\alpha =-2\Rightarrow$no solution.