The system of equations

$αx + y + z = a -1$

$x+αy +z = α-1$

$x+y+αz=α-1$

has no solution, if $α$ is

- 2

For $α = 1$, the system reduces to a homogeneous system which is always consistent. So, $α ≠1$.

For $α ≠1$, we have

$D=\begin{vmatrix}α&1&1\\1&α&1\\1&1&α\end{vmatrix}=\begin{vmatrix}α+2&α+2&α+2\\1&α&1\\1&1&α\end{vmatrix}$  [Applying $R_1→ R_1+ R_2+ R_3$]

$⇒D=(α+2)\begin{vmatrix}1&1&1\\1&α&1\\1&1&α\end{vmatrix}=(α+2)\begin{vmatrix}1&0&0\\1&α-1&0\\1&0&α-1\end{vmatrix}$  [Applying $C_2 →C_2-C_1, C_3 →C_3-C_1$]

$⇒D=(α+2)(α-1)^2$

and, $D_1=\begin{vmatrix}α-1&1&1\\α-1&α&1\\α-1&1&α\end{vmatrix}=(α-1)\begin{vmatrix}1&1&1\\1&α&1\\1&1&α\end{vmatrix}$

$⇒D_1(α-1)\begin{vmatrix}1&0&0\\1&α-1&0\\1&0&α-1\end{vmatrix}$ [Applying $C_2 →C_2-C_1, C_3→C_3-C_1$]

$⇒D_1(α-1)^3$

Clearly, $D = 0$ for $α = -2$ but $D_1 ≠ 0$.

So, the system is inconsistent for $α = -2$.