Question

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

Answer 1

We have,

$\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$

Solving that,

$=1-c\omega -a(\omega -{\omega }^{2}c)$

$=1(1-c\omega )-a\omega (1-c\omega )$

$=(1-c\omega )(1-a\omega )$

For non singular matrix.

$c\ne \frac{1}{\omega }anda\ne \frac{1}{\omega }$

$\Rightarrow c\ne {\omega }^2,a\ne {\omega }^2$

A and c must be $\omega$and b can be $\omega or{\omega }^2$.

Then, total matrix $=2$.

Hence, this is the answer.