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)

2

For being non-singular
$\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]\ne 0$
$\Rightarrow ac{\omega }^2-(a+c)\omega +1\ne 0$
hence number of possible triplets of (a , b ,c ) is 2 .
$i.e.(\omega ,{\omega }^2,\omega )$ and $(\omega ,\omega ,\omega )$