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Q.
If $x=c y+b z, y=a z+c x, z=b x+a y$ where $x, y, z$ are not all zeros, then the value of $a^{2}+b^{2}+c^{2}+2 a b c$ is
Determinants
Solution:
Given system $x-c y-b z=0, c x-y+a z=0, b x+a y-z=0$ System has non-trivial solution then
$\begin{vmatrix}1&-c&-b\\ c&-1&a\\ b&a&-1\end{vmatrix}=0 $
$\Rightarrow 1-a b c-a b c-b^{2}-a^{2}-c^{2}=0$
$\Rightarrow a^{2}+b^{2}+c^{2}+2 a b c=1$