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

Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form 1 a b ω 1 c ω² ω 1 where each of a, b and c is either ω or ω². Then the number of distinct matrices in the set S is $\omega^{3} = 1, \quad \omega \neq 1, \quad 1 + \omega + \omega^{2} = 0$ $A = \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^{2} & \omega & 1 \end{bmatrix}, \quad a, b, c \in \{\omega, \omega^{2}\}$ $\text{Count } A \text{ with } \det A \neq 0, \text{ among the } 2^{3} = 8 \text{ choices}$ $\det A = 1(1 - c\omega) - a(\omega - c\omega^{2}) + b(\omega^{2} - \omega^{2})$ $\omega^{2} - \omega^{2} = 0 \Rightarrow \text{the } b \text{ term vanishes}$ $\det A = 1 - c\omega - a\omega + ac\,\omega^{2}$ $\det A = 1 - \omega(a+c) + \omega^{2} ac$ $(a,c)=(\omega,\omega):\ \det = 1 - 2\omega^{2} + \omega^{4} = 1 + \omega - 2\omega^{2}$ $= -3\omega^{2} \neq 0 \ \Rightarrow \text{non-singular}$ $(a,c)=(\omega,\omega^{2}) \text{ or } (\omega^{2},\omega):\ \det = 1 + \omega + \omega^{2} = 0$ $(a,c)=(\omega^{2},\omega^{2}):\ \det = 1 - 2\omega^{3} + \omega^{6} = 1 - 2 + 1 = 0$ $\text{Only } (a,c) = (\omega,\omega) \text{ gives } \det \neq 0$ $b \in \{\omega, \omega^{2}\}:\ 2 \text{ choices}, \qquad (a,c) = (\omega,\omega) \text{ fixed}$ $\text{Number of non-singular matrices} = 2$ $\boxed{\text{Number of matrices in } S = 2} \ \Rightarrow \text{option (1)}$ $\det = 1 - \omega(a+c) + \omega^{2}ac; \quad \text{only } a = c = \omega; \quad 2 \text{ matrices}$