Question

If $\omega$ is the complex cube root of unity, then prove that $\left|\begin{array}{ccc}1& 1& 1\\ 1& -1-{\omega }^2& {\omega }^2\\ 1& {\omega }^2& {\omega }^4\end{array}\right|=\pm 2\sqrt{3}i$.

Answer 1

$\left|\begin{array}{ccc}1& 1& 1\\ 1& -1-w^2& w^2\\ 1& w^2& w^4\end{array}\right|$

$=1((-1-w^2)w^4-w^4)-1(w^2-w^4)+1(w^2+1+w^2)$

$=-w^4-w^6-w^4-w^2+w^4+2w^2+1$

$=-w-1-w-w^2+w+2w^2+1$

$=-w+w^2$

$\Rightarrow w=\frac{-1\pm \sqrt{3}i}{2}$

$=\frac{1}{2}\mp \sqrt{3}i-\frac{1}{2}\mp \sqrt{3}i$

$=\pm 2\sqrt{3}i$

Hence, the answer is $\pm 2\sqrt{3}i.$