$If a + b + c = 0, then the solution of the equation$ $|\begin{matrix} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{matrix}| = 0 is$

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$0, \pm \sqrt{(a^{2} + b^{2} + c^{2})}$

$0, \pm \sqrt{\frac{3}{2} (a^{2} + b^{2} + c^{2})}$

$\pm \frac{3}{2} (a^{2} + b^{2} + c^{2})$

answer is C.

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Detailed Solution

$\begin{array}{l} |\begin{matrix} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{matrix}| = 0 \\ \Rightarrow |\begin{matrix} a + b + c - x & c & b \\ a + b + c - x & b - x & a \\ a + b + c - x & a & c - x \end{matrix}| = 0 \\ \Rightarrow (Σ a - x) |\begin{matrix} 1 & c & b \\ 1 & b - x & a \\ 1 & a & c - x \end{matrix}| = 0 \\ \Rightarrow x = Σ a = 0 \\ \{(b - x) (c - x) - a^{2}\} - c \{c - x - a_{}}^{t} + b {a - b + x} = 0 \\ o r x^{2} - (a^{2} + b^{2} + c^{2}) + (a b + b c + c a) = 0 \\ x^{2} - Σ a^{2} + Σ a b = 0 \\ x^{2} - (Σ a^{2}) - \frac{1}{2} (Σ a^{2}) = 0 \\ [∵ a + b + c = 0 \Rightarrow {(a + b + c)}^{2} = 0 \\ \Rightarrow Σ a^{2} + 2 Σ a b = 0 \Rightarrow Σ a b = - \frac{1}{2} Σ a^{2}] \\ O r x = \pm \sqrt{\frac{3}{2} Σ a^{2}} \\ The solution is x = 0 o r \pm \sqrt{\frac{3}{2} Σ a^{2}} \end{array}$