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 the given matrix to be non-singular

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

$(1-aw)(1-cw)\ne 0$

$a\ne w^2$ and $c\ne w^2$

where $w$ is complex cube root of unity

And $a,b,c$ are complex cube root of unity

therefore, $a$ and $c$ can take two values $w$ and $w^2$

Hence, total numbers of distinct matrices

$=1\times 1\times 2=2$