Q. Let $ \omega \ne 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

Solution:

| A | $\ne $ 0, as non-singular
$\therefore \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin {bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end {bmatrix} \ne 0$
$\Rightarrow \ \ \ \ \ 1(1-c\omega)-a(\omega-\omega^2)+b(\omega^2+\omega^2) \ne 0$
$\Rightarrow \ \ \ \ \ 1-c\omega-a\omega+ac\omega^2 \ne 0 \ \ \Rightarrow \ \ \ (1-c\omega) (1-a\omega)\ne 0$
$\Rightarrow \ \ \ \ a \ne \frac{1}{\omega}, c \ne \frac{1}{\omega}$
$\Rightarrow \ \ \ \ a=\omega, c=\omega $ and $ b \ \in \{\omega,\omega^2\} \ \ \Rightarrow \ \ \ 2$ solutions

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