Question

If $\omega$ is a non-real cube root of unity and n is not a multiple of 3, then $\Delta =\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^{2n}& 1& {\omega }^n\\ {\omega }^n& {\omega }^{2n}& 1\end{array}\right|$ is equal to

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

Answer 1

The correct option is A $0$

$\Delta =\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^{2n}& 1& {\omega }^n\\ {\omega }^n& {\omega }^{2n}& 1\end{array}\right|$
$=1(1-{\omega }^{3n})-{\omega }^n({\omega }^{2n}-{\omega }^{2n})+{\omega }^{2n}({\omega }^{4n}-{\omega }^n)$
$=0$