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

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