<s>[INST] Given $3$ equations with parameters $a, b, c$, evaluate $a^2+b^2+c^2$ 
Given the equations 
$$\begin{align}
x&=cy+bz \\
y&=az+cx \\
z&=bx+ay 
\end {align}$$
where $x,y,z$ are not all zero, evaluate $a^2+b^2+c^2$.

I don't understand the way to relate $a,b,c$ and $x,y,z$.
Can someone explain this?
 [/INST] We can rearrange the equations as $-x+cy+bz=0$, $cx-y+az=0$, $bx+ay-z=0$, and so
$$\begin{pmatrix}-1&c&b\\c&-1&a\\b&a&-1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}.$$
This has a solution with $x,y,z$ not all zero if and only if the determinant
$$\begin{vmatrix}-1&c&b\\c&-1&a\\b&a&-1\end{vmatrix}=0.$$
</s>