Complex Numbers

Let $\omega$ be a complex number such that 2$\omega$ + 1 = z where z = $\sqrt {-3}$. If \left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k, then k is equal to :