Question

The value of $\left| \begin{matrix} 1 & \omega & 2\omega^{2} \\ 2 & 2\omega^{2} & 4\omega^{3} \\ 3 & 3\omega^{3} & 6\omega^{4} \end{matrix} \right|$is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

0

Answer 2

Sol.

1 & \omega & 2\omega^{2} \\ 2 & 2\omega^{2} & 4\omega^{3} \\ 3 & 3\omega^{3} & 6\omega^{4} \end{matrix} \right| = 2\left| \begin{matrix} 1 & \omega & \omega^{2} \\ 2 & 2\omega^{2} & 2 \\ 3 & 3 & 3\omega \end{matrix} \right|$$ = 2[1(6w³ – 6) – w (6 w – 6) + w ² (6 – 6 w ²)] = 12[–w(w – 1) + w²(1 – w²)] ̃ 12[–w² + w + w² – w⁴]= 12[–w ² – 1 – w] = – 12[1 + w + w ²] = 0