Given, the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha ,$ and $-\alpha x+y+2z=-\alpha$ 

Now for system to be inconsistent, $∆=0 \& \text{either of } {∆}_1,{∆}_2 \text{or} {∆}_3 \ne 0$

So, $\Delta =\left|\begin{array}{ccc}1& 2& 1\\ 2& 3& -1\\ -2& 1& 2\end{array}\right|$

$=\left(6+\mathrm{y}\right)-2\left(2\alpha -\alpha \right)+1\left(\alpha +3\alpha \right)$

$=7-2\alpha +4\alpha$

$=7+2\alpha$

$\Delta =0\Rightarrow \alpha =-\frac{7}{2}$

Now calculating, ${\Delta }_1=\left|\begin{array}{ccc}2& 2& 1\\ \alpha & 3& -1\\ -\alpha & 1& 2\end{array}\right|$

$=14+2\alpha$

Where $\alpha =-\frac{7}{2}$ so, ${∆}_1=14-7=7$

${\Delta }_1\ne 0$, hence system is inconsistent at $\alpha =\frac{-7}{2}$