JEE Main 2013 — Hyperbola Question with Solution

Given system of equations is homogeneous which is $$\begin{array}{rl}& x+ay=0\\ & y+az=0\\ & z+ax=0\end{array}$$ It can be written in matrix form as $$\mathrm{A}=\left(\begin{array}{lll}1& a& 0\\ 0& 1& a\\ a& 0& 1\end{array}\right)$$ Now, $|\mathrm{A}|=\left[1-a\left(-a^2\right)\right]=1+a^3\ne 0$ So, system has only trivial solution. Now, $|\mathrm{A}|=0$ 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 solution. Hence set of all real values of ' $a$ ' is $$\mathrm{R}-\{-1\}$$