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Problem 32: Evalaute

\begin{align*} \Delta =\left| \begin{array}{ccc} 1& \omega& {\omega}^2\\ \omega& {\omega}^2& 1\\ {\omega}^2& 1& \omega\\ \end{array}\right| \end{align*} where $\omega$ is one of the imaginary cube roots of unity.

Problem 32: Evalaute

\begin{align*} \Delta =\left| \begin{array}{ccc} 1& \omega& {\omega}^2\\ \omega& {\omega}^2& 1\\ {\omega}^2& 1& \omega\\ \end{array}\right| \end{align*} where $\omega$ is one of the imaginary cube roots of unity.

Solution:

Applying $C_1\rightarrow C_1+C_2+C_3$

\begin{align*} \Delta =\left| \begin{array}{ccc} 1+\omega+{\omega}^2& \omega& {\omega}^2\\ 1+\omega+{\omega}^2& {\omega}^2& 1\\ 1+\omega+{\omega}^2& 1& \omega\\ \end{array}\right| \end{align*} We know that $1+\omega+{\omega}^2 =0$, where $\omega$ is one of the imaginary cube roots of unity. \begin{align*} \Delta =\left| \begin{array}{ccc} 0& \omega& {\omega}^2\\ 0& {\omega}^2& 1\\ 0& 1& \omega\\ \end{array}\right|\\ \Delta =0. \end{align*}

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