The number of real values of ‘a’ for which the system & equations $x+ay-z=0,2x-y+az=0,ax+y+2z=0$ has a non-trivial solution is

$The condition for a homogeneous system to have a non-trivial solution is \Delta =0$

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

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

$\Rightarrow \left(2+a\right)\left\{0\left(-2+a\right)-a\left(2-a\right)-1\left(1+1\right)\right\}=0$

$\Rightarrow 2+a=0,-2a+a^2-2=0\Rightarrow a^2-2a-2=0\Rightarrow a=1\mp \sqrt{3}$