Let $\omega=-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}, \mathrm{i}=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & 1 & 1 \ 1 & -1-\omega^2 & \omega^2 \ 1 & \omega^2 & \omega^4\end{array} ight|$ is

Question

Let $\omega=-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}, \mathrm{i}=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & 1 & 1 \ 1 & -1-\omega^2 & \omega^2 \ 1 & \omega^2 & \omega^4\end{array}ight|$ is

Aicharya · Accepted Answer

$\begin{aligned} & \quad \omega^2=\left(-\frac{1}{2}ight)^2+\left(\mathrm{i} \frac{\sqrt{3}}{2}ight)^2-2 \mathrm{i}\left(\frac{\sqrt{3}}{4}ight)=-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i} \ & \quad \omega^3=\frac{1}{4}+\frac{3}{4}=1 \ & 	herefore \quad 1+\omega+\omega^2=0\end{aligned}$ $\begin{aligned} & 	herefore \quad\left|\begin{array}{ccc}1 & 1 & 1 \ 1 & -1-\omega^2 & \omega^2 \ 1 & \omega^2 & \omega^4\end{array}ight|=\left|\begin{array}{ccc}1 & 1 & 1 \ 1 & \omega & \omega^2 \ 1 & \omega^2 & \omega\end{array}ight| \ & \quad=1\left(\omega^2-\omega^4ight)-1\left(\omega-\omega^2ight)+1\left(\omega^2-\omegaight) \ & \quad=\omega^2-\omega^4-\omega+\omega^2+\omega^2-\omega \ & \quad=-\omega^4+3 \omega^2-2 \omega \ & \quad=-\omega+3 \omega^2-2 \omega \ & =3 \omega^2-3 \omega \ & \quad=3 \omega(\omega-1)\end{aligned}$

Let $\omega=-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}, \mathrm{i}=\sqrt{-1}$, then the value of…

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