Search any question & find its solution
Question:
Answered & Verified by Expert
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 $S$ is
Options:
Solution:
2007 Upvotes
Verified Answer
The correct answer is:
2
2
$|A| \neq 0$, as non-singular.
$$
\begin{aligned}
& \therefore \quad\left|\begin{array}{ccc}
1 & a & b \\
\omega & 1 & c \\
\omega^2 & \omega & 1
\end{array}\right| \neq 0 \\
& \Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^2\right) \\
& +b\left(\omega^2-\omega^2\right) \neq 0 \\
& \Rightarrow \quad 1-c \omega-a \omega+a c \omega^2 \neq 0 \\
&
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow \quad(1-c \omega)(1-a \omega) \neq 0 \\
& \Rightarrow \quad a \neq \frac{1}{\omega}, c \neq \frac{1}{\omega} \Rightarrow a=\omega, c=\omega
\end{aligned}
$$
and $b \in\left\{\omega, \omega^2\right\} \Rightarrow 2$ solutions
$$
\begin{aligned}
& \therefore \quad\left|\begin{array}{ccc}
1 & a & b \\
\omega & 1 & c \\
\omega^2 & \omega & 1
\end{array}\right| \neq 0 \\
& \Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^2\right) \\
& +b\left(\omega^2-\omega^2\right) \neq 0 \\
& \Rightarrow \quad 1-c \omega-a \omega+a c \omega^2 \neq 0 \\
&
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow \quad(1-c \omega)(1-a \omega) \neq 0 \\
& \Rightarrow \quad a \neq \frac{1}{\omega}, c \neq \frac{1}{\omega} \Rightarrow a=\omega, c=\omega
\end{aligned}
$$
and $b \in\left\{\omega, \omega^2\right\} \Rightarrow 2$ solutions
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.