Question

If $\omega$ be a complex cube root of unity, then $\left|\begin{array}{ccc}1& \omega & -\frac{{\omega }^2}{2}\\ 1& 1& 1\\ 1& -1& 0\end{array}\right|=$

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

Answer 1

$\left|\begin{array}{ccc}1& \omega & -\frac{{\omega }^2}{2}\\ 1& 1& 1\\ 1& -1& 0\end{array}\right|=-\frac{1}{2}\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 1& 1& -2\\ 1& -1& 0\end{array}\right|=-\frac{1}{2}\left|\begin{array}{ccc}0& \omega & {\omega }^2\\ 0& 1& -2\\ 0& -1& 0\end{array}\right|=0,(Apply{C}_1\to C_1+C_2+C_3)$