If $w=\frac{-1-i \sqrt{3}}{2}$ where $i=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & w & w^2 \ w & w^2 & 1 \ w^2 & 1 & w\end{array} ight|$ is

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If $w=\frac{-1-i \sqrt{3}}{2}$ where $i=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & w & w^2 \ w & w^2 & 1 \ w^2 & 1 & w\end{array}ight|$ is

Aicharya · Accepted Answer

$\begin{aligned} & \omega=\frac{-1-\mathrm{i} \sqrt{3}}{2}, \omega^2=\frac{1-3+2 \mathrm{i} \sqrt{3}}{4}=\frac{-1+\mathrm{i} \sqrt{3}}{2}, \ & \omega^3=1, \omega^6=1 \ & \left|\begin{array}{ccc}1 & \omega & \omega^2 \ \omega & \omega^2 & 1 \ \omega^2 & 1 & \omega\end{array}ight| \ & =1\left(\omega^3-1ight)-\omega\left(\omega^2-\omega^2ight)+\omega^2\left(\omega-\omega^4ight) \ & =1\left(\omega^3-1ight)+0+\omega^2\left(\omega-\omega^4ight) .\end{aligned}$ $\begin{aligned} & =\omega^3-1+\omega^3-\omega^6 \ & =2 \omega^3-\omega^6-1 \ & =2(1)-1-1 \ & =0\end{aligned}$

If $w=\frac{-1-i \sqrt{3}}{2}$ where $i=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & w & w^2…

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