Question

If $\omega$ is the cube root of unity, then $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|=$

• A. $1$
• B. $0$
• C. $\omega$
• D. ${\omega }^2$

Answer 1

The correct option is B $0$

$\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|=\left|\begin{array}{ccc}1+\omega +{\omega }^2& \omega & {\omega }^2\\ 1+\omega +{\omega }^2& {\omega }^2& 1\\ 1+\omega +{\omega }^2& 1& \omega \end{array}\right|=\left|\begin{array}{ccc}0& \omega & {\omega }^2\\ 0& {\omega }^2& 1\\ 0& 1& \omega \end{array}\right|=0.$