SATHEE: Matrices and Determinants 3 Question 8

Matrices and Determinants 3 Question 8

10. Let $ω \neq 1$ be a cube root of unity and $S$ be the set of all, where non-singular matrices of the form

$[\begin{matrix} 1 & a & b \\ ω & 1 & c, \\ ω^{2} & ω & 1 \end{matrix}]$ , where each of $a, b$ and $c$ is either $ω$ or $ω^{2}$ . Then, the number of distinct matrices in the set $S$ is

(2011)

(a) 2

(b) 6

(d) 8

Show Answer

Answer:

Correct Answer: 10. $(a)$

Solution:

$| A | \neq 0$ , as non-singular $[\begin{matrix} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{matrix}] \neq 0$

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

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

$\Rightarrow (1 - c ω) (1 - a ω) \neq 0 \Rightarrow a \neq \frac{1}{ω}, c \neq \frac{1}{ω}$

$\Rightarrow a = ω, c = ω$ and $b \in (ω, ω^{2}) \Rightarrow 2$ solutions

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Matrices and Determinants 3 Question 8

10. Let ω≠1 be a cube root of unity and S be the set of all, where non-singular matrices of the form

Answer:

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10. Let $ω \neq 1$ be a cube root of unity and $S$ be the set of all, where non-singular matrices of the form