Key Concepts and Formulas

This problem primarily relies on the properties of the cube roots of unity and standard matrix multiplication.

1. Cube Roots of Unity: The roots of the equation $x^3 - 1 = 0$ are $1, \omega, \omega^2$. The equation $x^2+x+1=0$ is derived from factoring $x^3-1=(x-1)(x^2+x+1)=0$, so its roots are $\omega$ and $\omega^2$. Key properties include:
   • $1 + \omega + \omega^2 = 0$
   • $\omega^3 = 1$
   • From $\omega^3=1$, it