Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form [ array ccc 1 & a & b ω & 1 & c ω^2 & ω & 1 array ] , where each of a, b and c is either ω or ω^2 . Then, the number of

Correct answer: A. 2 |A| ≠ 0 , as non-singular. aligned & | array ccc 1 & a & b ω& 1 & c ω^2 & ω& 1 array | ≠ 0 & 1(1-c ω)-a(ω-c ω^2) & +b (ω^2-ω^2) ≠ 0 & 1-c ω-a ω+a c ω^2 ≠ 0 & aligned aligned & (1-c ω)(1-a ω) ≠0 & a ≠ (1)/(ω),c ≠ (1)/(ω) =ω,c=ω aligned and b ∈ \ ω, ω^2\ 2 solutions

JEE Advanced 2011Paper 2

Exam: JEE Advanced
Year: 2011
Paper: Paper 2
Subject: Mathematics
Chapter: Matrices

JEE Advanced 2011 · Paper 2 · Mathematics · Matrices

Let ω ≠ 1 be a cube root of unity and S be the set of all...

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

\omega^2

. Then, the number of distinct matrices in the set

S

Options

Correct answer

A. 2

Step-by-step solution

|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| ≠ 0 \\ & \Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^2\right) \\ & +b \left(\omega^2-\omega^2\right) ≠ 0 \\ & \Rightarrow \quad 1-c \omega-a \omega+a c \omega^2 ≠ 0 \\ & \end{aligned}

\begin{aligned}& \Rightarrow \quad(1-c \omega)(1-a \omega) \neq0 \\ & \Rightarrow \quad a \neq \frac{1}{\omega},c \neq \frac{1}{\omega} \Rightarrowa=\omega,c=\omega \end{aligned}

and

b \in \left\{\omega, \omega^2\right\}

\Rightarrow 2$ solutions

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