Question

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

• A. $2$
• B. $6$
• C. $4$`
• D. $8$

Answer 1

The correct option is A $2$

Since, the given matrix $\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$ is non-singular.

So, $\Delta =\left|\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right|\ne 0$

$1(1-\omega c)-a(\omega -{\omega }^{2}c)+b({\omega }^2-{\omega }^2)\ne 0$

$1-\omega (a+c){+ac\omega }^2\ne 0$

$a+c\ne -1$ ......(1)

and $ac\ne 1$ ......(2)

So, $a=\omega$ or ${\omega }^2$ and $c=\omega$ or ${\omega }^2$

If $c={\omega }^2$, then $\Delta =0$. So $c\ne {\omega }^2$

So, $c=\omega$

So, by equation (1), $a\ne {\omega }^2$.

Hence, $a=\omega$.

Since, the determinant value is independent of $b$ . So $b$ can be $\omega$ or ${\omega }^2$.

Hence, $c=\omega ,a=\omega ,b=\omega or{\omega }^2$.

So, number of matrices formed will be $2$.