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

$$\begin{gathered} The condition for non-exis\tan ce of solution is \Delta =0 \\ \Rightarrow \left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \alpha & 1\\ 1& 1& \alpha \end{array}\right|=0 \end{gathered}$$

$\Rightarrow \alpha ({\alpha }^2-1)-1(\alpha -1)+1(1-\alpha )=0$

$\Rightarrow {\alpha }^3-3\alpha +2=0\Rightarrow \alpha =-2,1$

  If $\alpha =1,$all are coincident 

It has infinitely many solutions 

 $\therefore \alpha =-2$