Matrices And Determinants

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