What is the reasoning behind the solution of the determinant of this matrix? What rules were used here?

Question

I can't figure out how this determinant was calculated. I tried using Sarrus' rule but ended up with a complicated expression.

Why do I make all the elements in my first column to $1+\omega +{\omega}^2$ ?

How do I get my answer as $0$ ?

Answer 1

The second and third columns were added to the first column. I guess that $\omega$ is the cube-root of $1$, so that makes the first column entries all $0$, which makes it easy to see that the determinant is $0$.

Answer 2

1, $\omega$ and $\omega ^2$ are the cube roots of unity. Sum of cube roots of unity is 0. So the determinant is 0