Question

Let $ \omega \ne 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin {bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end {bmatrix}$ 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)

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