Explanation

(c)
Given, $\left| \begin{matrix}
a - x & c & b \\
c & b - x & a \\
b & a & c - x
\end{matrix} \right| = 0$
Applying $R_{1} \rightarrow R_{1} + R_{2} + R_{3}$
$\Rightarrow (a + b + c - x)\left| \begin{matrix}
1 & 1 & 1 \\
c & b - x & a \\
b & a & c - x
\end{matrix} \right| = 0$
$\Rightarrow \ (a + b + c - x)\left| \begin{matrix}
1 & 0 & 0 \\
c & b - x - c & a - c \\
b & a - b & c - x - b
\end{matrix} \right| = 0$
$\Rightarrow \ (a + b + c - x)\left\lbrack 1(b - x - c)(c - x - b) - (a - c)(a - b) \right\rbrack = 0$
$\Rightarrow \ (a + b + c - x)\left\lbrack bc - xb - b^{2} - xc + x^{2} + bx - c^{2} + cx + bc - \left( a^{2} - ab - ac + bc \right) \right\rbrack = 0$
$\Rightarrow \ (a + b + c - x)\left\lbrack x^{2} - a^{2} - b^{2} - c^{2} + ab + bc + ca \right\rbrack = 0$
$\Rightarrow x = a + b + c\ or\ x^{2} = a^{2} + b^{2} + c^{2} + ab + bc + ca$
$\Rightarrow x = 0\ or\ x^{2} = a^{2} + b^{2} + c^{2} + \frac{1}{2}(a^{2} + b^{2} + c^{2})$
$\Rightarrow x = 0\ or\ x = \pm \sqrt{\frac{3}{2}(a^{2} + b^{2} + c^{2})}\$