Explanation

(a)
Given,$\ \left| \begin{matrix}
bc & ca & ab \\
ca & ab & bc \\
ab & bc & ca
\end{matrix} \right| = 0$
$\Rightarrow (ab)^{3} + (bc)^{3} + (ca)^{3} - 3a^{2}b^{2}c^{2} = 0$
$\Rightarrow (ab + bc + ca)(a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2} - ab^{2}c$
$- bc^{2}a - ca^{2}b) = 0$
$\Rightarrow \ \ \ ab + bc + ca = 0$
$\Rightarrow \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$