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Determinant

If the system...

Question

If the system of equations $x = c y + b z, y = a z + c x a n d z = b x + a y$ has a non-trival solution, then value of $a^{2} + b^{2} + c^{2} + 2 a b c$ is

A

1

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B

0

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C

- 1

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D

independent of

a, b, c

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Solution

The correct options are
A independent of $a, b, c$
D $1$
Equation will have no trivial solution when
$∣ ∣ ∣ ∣ \begin{matrix} - 1 & c & b c & - 1 & a b & a & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow (- 1) (1 - a^{2}) - c (- c - a b) + b (a c + b) = 0 \Rightarrow a^{2} + b^{2} + c^{2} + 2 a b c = 1$

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