Matrices and Determinants 3 Question 8
10. Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all, where 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
(2011)
(a) 2
(b) 6
(c) 4
(d) 8
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Answer:
Correct Answer: 10. $(a)$
Solution:
- $|A| \neq 0$, as non-singular $\begin{bmatrix}1 & a & b \\ \omega & 1 & c \\ \omega^{2} & \omega & 1\end{bmatrix} \neq 0$
$\Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^{2}\right)+b\left(\omega^{2}-\omega^{2}\right) \neq 0$
$\Rightarrow \quad 1-c \omega-a \omega+a c \omega^{2} \neq 0$
$\Rightarrow \quad(1-c \omega)(1-a \omega) \neq 0 \Rightarrow a \neq \frac{1}{\omega}, c \neq \frac{1}{\omega}$
$\Rightarrow \quad a=\omega, c=\omega$ and $b \in(\omega, \omega^{2}) \quad \Rightarrow \quad 2$ solutions
