If 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, when $\alpha$ is

$\left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \alpha & 1\\ 1& 1& \alpha \end{array}\right|=0\Rightarrow \alpha =1,\alpha =-2$
But for $\alpha$=1, clearly there are infinitely many solutions and when we put $\alpha$=-2 in given system equations has no solution.