$\text{ The value of }\left|\mathrm{\alpha }\right|\text{ for which the system of equation }$
$\begin{array}{r}\mathrm{\alpha x}+\mathrm{y}+\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}+\mathrm{\alpha y}+\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}+\mathrm{y}+\mathrm{\alpha z}=\mathrm{\alpha }-1\end{array}\mathrm{has} \mathrm{no} \mathrm{solution}, \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

System of equations
$\begin{array}{r}\mathrm{\alpha x}+\mathrm{y}+\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}+\mathrm{\alpha y}+\mathrm{z}=\mathrm{\alpha }-1\\ \mathrm{x}+\mathrm{y}+\mathrm{\alpha z}=\mathrm{\alpha }-1\end{array}$
Since system has no solution.
$\text{ Therefore, (1) }\mathrm{\Delta }=0\text{ and (2) }\mathrm{\alpha }-1\ne 0$
$\begin{array}{l}\left|\begin{array}{ccc}\mathrm{\alpha }& 1& 1\\ 1& \mathrm{\alpha }& 1\\ 1& 1& \mathrm{\alpha }\end{array}\right|=0,\mathrm{\alpha }\ne 1\\ {\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_3,{\mathrm{R}}_2\to {\mathrm{R}}_2\to {\mathrm{R}}_3\\ \left|\begin{array}{ccc}\mathrm{\alpha }-1& 0& 1-\mathrm{\alpha }\\ 0& \mathrm{\alpha }-1& 1-\mathrm{\alpha }\\ 1& 1& \mathrm{\alpha }\end{array}\right|=0\\ \mathrm{\alpha }=1,1,-2\text{ or }\mathrm{\alpha }=1,-2\end{array}$