Question

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

• A. 1
• B. -1
• C. None of these

Answer 1

$\Delta \equiv \left|\begin{array}{ccc}2& 2\omega & -{\omega }^2\\ 1& 1& 1\\ 1& -1& 0\end{array}\right|=\left|\begin{array}{ccc}2+2\omega +2{\omega }^2& 2\omega & -{\omega }^2\\ 1+1-2& 1& 1\\ 1-1-0& -1& 0\end{array}\right|(C_1\to C_1+C_2-2C_3)=\left|\begin{array}{ccc}0& 2\omega & -{\omega }^2\\ 0& 1& 1\\ 0& -1& 0\end{array}\right|=0$.