The system of equations $\begin{array}{c}\mathrm{\alpha x}-\mathrm{y}-\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}-\mathrm{\alpha y}-\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}-\mathrm{y}-\mathrm{\alpha z}=\mathrm{\alpha }-1\end{array}$has no solution if $\mathrm{\alpha }$ is

For no solution or infinity many solutions $\left|\begin{array}{ccc}\mathrm{\alpha }& -1& -1\\ 1& -\mathrm{\alpha }& -1\\ 1& -1& -\mathrm{\alpha }\end{array}\right|=0$

$$\begin{gathered} \mathrm{Or} \mathrm{\alpha }\left({\mathrm{\alpha }}^2-1\right)-1(\mathrm{\alpha }-1)+1(1-\mathrm{\alpha })=0 \\ \mathrm{Or} \mathrm{\alpha }\left({\mathrm{\alpha }}^2-1\right)-2\mathrm{\alpha }+2=0 \\ \mathrm{Or}\mathrm{\alpha }(\mathrm{\alpha }-1)(\mathrm{\alpha }+1)-2(\mathrm{\alpha }-1)=0 \\ \mathrm{Or}(\mathrm{\alpha }-1)\left({\mathrm{\alpha }}^2+\mathrm{\alpha }-2\right)=0 \\ \text{ Or }(\mathrm{\alpha }-1)(\mathrm{\alpha }+2)(\mathrm{\alpha }-1)=0 \\ \mathrm{Or}(\mathrm{\alpha }-1{)}^2(\mathrm{\alpha }+2)=0 \\ \text{ Or }\mathrm{\alpha }=1,1,-2 \\ If\alpha =1 \mathrm{the} \mathrm{system} \mathrm{ofequations} \mathrm{are} \mathrm{coincident} \mathrm{homogeneous} \mathrm{equations} \\ \mathrm{and} \mathrm{have} \mathrm{infinite} \mathrm{solutions} \\ \therefore \alpha =-2 \end{gathered}$$