Explanation

(d)
We have, $\mathrm{\Delta} \equiv \left| \begin{matrix}
a & b & 0 \\
0 & a & b \\
b & 0 & a
\end{matrix} \right| = 0$
$\Rightarrow \ \mathrm{\Delta} \equiv a\left( a^{2} - 0 \right) - b\left( 0 - b^{2} \right) = a^{3} + b^{3}$
$\Rightarrow \ a^{3} + b^{3} = 0\ \Rightarrow \left( \frac{a}{b} \right)^{3} = - 1$
$\therefore\left( \frac{a}{b} \right)$ is one of the cube roots of $- 1.$