The value of  $\left|\alpha \right|$ for which 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 is ________
 

The system of equations has no solution if

$\left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \alpha & 1\\ 1& 1& \alpha \end{array}\right|=0$

$$\begin{gathered} C_1\to C_1+C_2+C_3 \\ \left|\begin{array}{l}\alpha +2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ \alpha +2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ \alpha +2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \end{array}\right|=0 \end{gathered}$$

$\begin{array}{r}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(\alpha +2)(\alpha -1{)}^2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=1,-2\end{array}$

If $\alpha =1$ given equations become identical which have infinite solutions

$\mathrm{If} \mathrm{\alpha }=-2 \mathrm{the} \mathrm{given} \mathrm{system} \mathrm{of} \mathrm{equations} \mathrm{has} \mathrm{no} \mathrm{solution}$