Let $\omega \neq 1$ be a cube root of unity and S be the set of all non-singular matrices of the form $\left[\begin{array}{ccc}1 & \mathrm{a} & \mathrm{b} \\ \omega & 1 & \mathrm{c} \\ \omega^2 & \omega & 1\end{array}\right]$, where each of $a, b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is