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

Let \(A=\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\) For non-singular matrix
\(
\begin{aligned}
& |\mathrm{A}| \neq 0 \\
& \Rightarrow\left|\begin{array}{ccc}
1 & \mathrm{a} & \mathrm{b} \\
\omega & 1 & \mathrm{c} \\
\omega^2 & \omega & 1
\end{array}\right| \neq 0
\end{aligned}
\)
\(\Rightarrow 1(1-\omega c)-a\left(\omega-\omega^2 c\right)+b(0) \neq 0 \)
\( \Rightarrow 1(1-\omega c)-a \omega(1-\omega c) \neq 0 \)
\( \Rightarrow(1-\omega c)(1-a \omega) \neq 0 \)
\( \Rightarrow c \neq \frac{1}{\omega} \text { and } a \neq \frac{1}{\omega} \)
\( \Rightarrow c \neq \omega^2 \text { and } a \neq \omega^2 \quad \ldots\left[\because \omega^3=1\right]\)
So possible value of a and \(\mathrm{c}\) is \(\omega\) only and \(\mathrm{b}\) can take values \(\omega\) or \(\omega^2\).
\(\therefore\) The possible number of distinct matrices \(=2\).