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

Answer: (A)

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