$\text{ If the equations }x+ay-z=0, 2x-y+az=0, \alpha x+y+2z=0\text{ have non - trivial solutions, }$$\text{ then }a\text{ can be }$

$\left|\begin{array}{ccc}1& a& -1\\ 2& -1& a\\ a& 1& 2\end{array}\right|=0$

$\text{ Applying }{R}_2\to R_2-2R_1\text{ and }{R}_3\to R_3-a{R}_1,\text{ we get }$

$$\begin{gathered} \left|\begin{array}{ccc}1& a& -1\\ 0& -1-2a& a+2\\ 0& 1-a^2& 2+a\end{array}\right|=0 \\ \Rightarrow -\left(1+2a\right)\left(a+2\right)-\left(a+2\right)\left(1-a^2\right)=0 \\ \Rightarrow \left(a+2\right)\left(a^2-2a-2\right)=0 \end{gathered}$$

$a=-2,1\pm \sqrt{3}$