Question

$\left|\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& w& 1\end{array}\right|$ Where w is a complex cube root of unity.

Answer 1

We know that

$w^2+w+1=0w^3=1$

Now, Expanding the given Determinant

$1(w^2-w)-w(w-w^2)+w^2(w^2-w^4)=w^2-w-w^2+w^3+w^4-w^6=-w+1+w^4-1^2=-w+w^4=-w+w.w^3=-w+w.1=-w+w=0$