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Let $\omega \neq 1$ 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 $\mathrm{S}$ is
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Verified Answer
The correct answer is:
$2$
Let $A=\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$ For non-singular matrix
$$
\begin{aligned}
& |\mathrm{A}| \neq 0 \\
& \Rightarrow\left|\begin{array}{ccc}
1 & \mathrm{a} & \mathrm{b} \\
\omega & 1 & \mathrm{c} \\
\omega^2 & \omega & 1
\end{array}\right| \neq 0
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow 1(1-\omega c)-a\left(\omega-\omega^2 c\right)+b(0) \neq 0 \\
& \Rightarrow 1(1-\omega c)-a \omega(1-\omega c) \neq 0 \\
& \Rightarrow(1-\omega c)(1-a \omega) \neq 0 \\
& \Rightarrow c \neq \frac{1}{\omega} \text { and } a \neq \frac{1}{\omega} \\
& \Rightarrow c \neq \omega^2 \text { and } a \neq \omega^2 \quad \ldots\left[\because \omega^3=1\right]
\end{aligned}
$$
So possible value of a and $\mathrm{c}$ is $\omega$ only and $\mathrm{b}$ can take values $\omega$ or $\omega^2$.
$\therefore \quad$ The possible number of distinct matrices $=2$.
$$
\begin{aligned}
& |\mathrm{A}| \neq 0 \\
& \Rightarrow\left|\begin{array}{ccc}
1 & \mathrm{a} & \mathrm{b} \\
\omega & 1 & \mathrm{c} \\
\omega^2 & \omega & 1
\end{array}\right| \neq 0
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow 1(1-\omega c)-a\left(\omega-\omega^2 c\right)+b(0) \neq 0 \\
& \Rightarrow 1(1-\omega c)-a \omega(1-\omega c) \neq 0 \\
& \Rightarrow(1-\omega c)(1-a \omega) \neq 0 \\
& \Rightarrow c \neq \frac{1}{\omega} \text { and } a \neq \frac{1}{\omega} \\
& \Rightarrow c \neq \omega^2 \text { and } a \neq \omega^2 \quad \ldots\left[\because \omega^3=1\right]
\end{aligned}
$$
So possible value of a and $\mathrm{c}$ is $\omega$ only and $\mathrm{b}$ can take values $\omega$ or $\omega^2$.
$\therefore \quad$ The possible number of distinct matrices $=2$.
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