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Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{lll}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$ where each of $a, b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is
$2$
$6$
$4$
$8$
Solution
$\left[\begin{array}{ccc} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{array}\right]$
Solving that,
$=1-c \omega-a\left(\omega-\omega^2 c\right)$
$=1(1-c \omega)-a \omega(1-c \omega)$
$=(1-c \omega)(1-a \omega)$
For non singular matrix.
$c \neq \frac{1}{\omega} \text { and } a \neq \frac{1}{\omega}$
$\Rightarrow c \neq \omega^2, \quad a \neq \omega^2$
A and $c$ must be $\omega$ and $b$ can be $\omega$ or $\omega^2$.
Then, total matrix $=2$.
Hence, this is the answer.
