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Question: \[\left| \begin{matrix} 1 & 1 & 1 \\ 1 & \omega^{2} & \omega \\ 1 & \omega & \omega^{2} \end{matrix}...

1 & 1 & 1 \\ 1 & \omega^{2} & \omega \\ 1 & \omega & \omega^{2} \end{matrix} \right| =$$
A

33i3\sqrt{3}i

B

−33i- 3\sqrt{3}i

C

i3i\sqrt{3}

D

3

Answer

33i3\sqrt{3}i

Explanation

Solution

∣1111ω2ω1ωω2∣=3(ω−ω2)\left| \begin{matrix} 1 & 1 & 1 \\ 1 & \omega^{2} & \omega \\ 1 & \omega & \omega^{2} \end{matrix} \right| = 3(\omega - \omega^{2})

=3[−1+3i2−−1−3i2]=33i\mathbf{= 3}\left\lbrack \frac{\mathbf{-}\mathbf{1 +}\sqrt{\mathbf{3}}\mathbf{i}}{\mathbf{2}}\mathbf{-}\frac{\mathbf{-}\mathbf{1}\mathbf{-}\sqrt{\mathbf{3}}\mathbf{i}}{\mathbf{2}} \right\rbrack\mathbf{= 3}\sqrt{\mathbf{3}}\mathbf{i}.