Let $\omega$ be the complex cube root of unity. Consider the set
$E=\left\{(a, b, c) \in \mathbb{R}^3: \operatorname{det}\left(\begin{array}{ccc}\omega & a & 1 \\ 0 & c & 1 \\ 1 & -b & \omega\end{array}\right)=0\right\}$
Then $E$ is

• A

  an empty set.
• B

  a nonempty finite set.
• C

  a plane in $\mathbb{R}^3$.
• Correct answer
  D

  a straight line in $\mathbb{R}^3$.

Hence, the correct answer is option: 'D'.