If $x = c y + b z, y = a z + c x, z = b x + a y$ , where $x, y, z$ are not all zero, then the value of $a^{2} + b^{2} + c^{2} + 2 a b c$

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Solution

The correct option is B $1$
Given equations are $x = c y + b z, y = a z + c x, z = b x + a y$
Given system has non-trivial solution
$∣ ∣ ∣ ∣ \begin{matrix} 1 & - c & - b c & - 1 & a b & a & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow 1 - a b c - a b c - b^{2} - a^{2} - c^{2} = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 a b c = 1$

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