Let $\omega
eq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{ccc}1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1\end{array} ight]$, where each of $a, b$ and $c$ is either $\omega$ or $\omega^2$. Then, the number of distinct…

Question

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

Aicharya · Accepted Answer

$|A| 
eq 0$, as non-singular. $ \begin{aligned} & 	herefore \quad\left|\begin{array}{ccc} 1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1 \end{array}ight| 
eq 0 \ & \Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^2ight) \ & +b\left(\omega^2-\omega^2ight) 
eq 0 \ & \Rightarrow \quad 1-c \omega-a \omega+a c \omega^2 
eq 0 \ & \end{aligned} $ $ \begin{aligned} & \Rightarrow \quad(1-c \omega)(1-a \omega) 
eq 0 \ & \Rightarrow \quad a 
eq \frac{1}{\omega}, c 
eq \frac{1}{\omega} \Rightarrow a=\omega, c=\omega \end{aligned} $ and $b \in\left\{\omega, \omega^2ight\} \Rightarrow 2$ solutions

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form…

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