Question

Let $\omega \ne 1$ be a cube root of unity and S be the set of all nonsingular 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

Let $\Delta =\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]=1(1-c\omega )-a(\omega -{\omega }^{2}c)+b(\omega \omega -{\omega }^2)=(1-a\omega )(1-c\omega )$
$\Delta$ is non singular $\iff \Delta \ne 0\iff a=c=\omega$
Note that b can take any vale $\omega$ or ${\omega }^2$
Thus there are two non singular matrices.
Hence, option 'A' is correct.