Given $3$ equations with parameters $a, b, c$, evaluate $a^2+b^2+c^2$

Question

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?

Answer 1

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 $$\begin{vmatrix}-1&c&b\\c&-1&a\\b&a&-1\end{vmatrix}=0.$$