Question

If $\omega \ne 1$is a cube root of unity and $S$ is the set of all non-singular matrices of the form if

$A=\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$

where each of $a,b,\mathrm{and}c$ is either $𝜔\mathrm{or}{𝜔}^2$. 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

$2$

As the given set is a set of all the non-singular matrices then the determinant of the given matrix cannot be equal to zero,

$\left|\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right|\ne 0$

Now, solve the determinant to determine the required,

$$\begin{gathered} \therefore 1\left(1-c\omega \right)-a\left(\omega -c{\omega }^2\right)+b\left({\omega }^2-{\omega }^2\right)\ne 0 \\ \Rightarrow 1\left(1-c\omega \right)-a\left(\omega -c{\omega }^2\right)\ne 0 \\ \Rightarrow 1\left(1-c\omega \right)-a\omega \left(1-c\omega \right)\ne 0 \\ \Rightarrow \left(1-a\omega \right)\left(1-c\omega \right)\ne 0 \end{gathered}$$

From this, we can conclude that,

$a\ne \frac{1}{\omega }\text{and}c\ne \frac{1}{\omega }$

Therefore, $a=\omega ,c=\omega ,\text{and}b$ belongs to $\left(\omega ,{\omega }^2\right)$

Hence, there can be two distinct matrices.