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Determinant

If x=cy+bz,...

Question

If $x = c y + b z$ , $y = c x + a z$ , $z = b x + a y$ has a non-trivial solution, then value of $a^{2} + b^{2} + c^{2} + 2 a b c$ equals to

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Solution

The correct option is C $1$
Given the system of equations has a non-trivial solution.
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} - 1 & c & b c & - 1 & a b & a & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow - (1 - a^{2}) - c (- c - a b) + b (a c + b) = 0$
$\Rightarrow - 1 + a^{2} + c^{2} + a b c + a b c + b^{2} = 0$
$∴ a^{2} + b^{2} + c^{2} + 2 a b c = 1$
Hence, option C.

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