$x + ay = 0$

 $y + az = 0$

$z + ax = 0$

It can be written inmatrix from as

$A = \left[ {\begin{array}{*{20}{c}}
1&a&0\\
0&1&a\\
a&0&1
\end{array}} \right]$

Now, $\left| A \right| = \left[ {1 - a\left( { - {a^2}} \right)} \right] = 1 + {a^3} \ne 0$

So,system has only trivial solution.

Now, $\left| A \right| = 0$ only when $a=-1$

So, system of equations has infinitely many solutions which is not possible because it is given that system has a unique solutioin.

Hence set of all real value of $'a'$ is

$R-{-1}$.