$$x + 2y + z = 2$$

$$\alpha x + 3y - z = \alpha $$

$$ - \alpha x + y + 2z = - \alpha $$

$$\Delta = \left| {\matrix{ 1 & 2 & 1 \cr \alpha & 3 & { - 1} \cr { - \alpha } & 1 & 2 \cr } } \right| = 1(6 + 1) - 2(2\alpha - \alpha ) + 1(\alpha + 3\alpha )$$

$$ = 7 + 2\alpha $$

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

$${\Delta _1} = \left| {\matrix{ 2 & 2 & 1 \cr \alpha & 3 & { - 1} \cr { - \alpha } & 1 & 2 \cr } } \right| = 14 + 2\alpha \ne 0$$ for $$\alpha = - {7 \over 2}$$

$$\therefore$$ For no solution $$\alpha = - {7 \over 2}$$