Explanation

(c)
Given, $\left| \begin{matrix}
a + x & a - x & a - x \\
a - x & a + x & a - x \\
a - x & a - x & a + x
\end{matrix} \right| = 0$
Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3}$ and taking common $(3a - x)$from $C_{1},$we get
$(3a - x)\left| \begin{matrix}
1 & a - x & a - x \\
1 & a + x & a - x \\
1 & a - x & a + x
\end{matrix} \right| = 0$
$\Rightarrow (3a - x)\left| \begin{matrix}
1 & a - x & a - x \\
0 & 2x & 0 \\
0 & 0 & 2x
\end{matrix} \right| = 0\left\lbrack \begin{array}{r}
R_{2} \rightarrow R_{2} - R_{1} \\
R_{3} \rightarrow R_{3} - R_{1}
\end{array} \right\rbrack$
$\Rightarrow$ $(3a - x)\left( 4x^{2} \right) = 0$
$\Rightarrow x = 3a,0$