Discussion Forum

Let $\omega  \neq 1$ 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

Answer:

Explanation:

$|A| \neq 0$, as non-singular

$\begin{bmatrix}1 & a&b \\\omega & 1&c\\\omega^{2}&\omega&1 \end{bmatrix} \neq=0$

$\Rightarrow$   $1(1-c \omega)-a(\omega-c \omega^{2})+b(\omega^{2}-\omega^{2}) \neq=0$

$\Rightarrow$  $1- c \omega-a \omega+ac \omega^{2} \neq 0$

 $\Rightarrow$   $(1-c \omega)(1-a \omega) \neq 0$

 $\Rightarrow$  $a\neq \frac{1}{\omega}, c \neq \frac{1}{\omega} \Rightarrow a= \omega , c= \omega$

 and    $b \epsilon( \omega , \omega^{2}) \Rightarrow 2$ solutions