SATHEE: Matrices and Determinants 1 Question 13

Matrices and Determinants 1 Question 13

13. Let $ω$ be a complex cube root of unity with $ω \neq 0$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j}$ . Then, $P^{2} \neq 0$ when $n$ is equal to

(2013 Adv.)

(a) 57

(b) 55

(d) 56

Show Answer

Answer:

Correct Answer: 13. (b,c,d)

Solution:

Here, $P = {[p_{i j}]}_{n \times n}$

with

$p_{i j} = w^{i + j}$

$∴$ When $n = 1$

$\begin{aligned} P & = {[p_{i j}]}_{1 \times 1} = [ω^{2}] \\ \Rightarrow P^{2} & = [ω^{4}] \neq 0 \end{aligned}$

$∴$ When $n = 2$

$P =$ ${[p_{i j}]}_{2 \times 2}$

= $[\begin{matrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{matrix}]$ = $[\begin{matrix} ω^{2} & ω^{3} \\ ω^{3} & ω^{4} \end{matrix}]$ = $[\begin{matrix} ω^{2} & 1 \\ 1 & ω \end{matrix}]$

$P^{2} = [\begin{matrix} ω^{2} & 1 \\ 1 & ω \end{matrix}]$ $[\begin{matrix} ω^{2} & 1 \\ 1 & ω \end{matrix}]$

$\Rightarrow P^{2} =$ $[\begin{matrix} ω^{4} + 1 & ω^{2} + ω \\ ω^{2} + ω & 1 + ω^{2} \end{matrix}]$ $\neq 0$

\omega^{2}

\omega

When $n = 3$

$P =$ ${[p_{i j}]}_{3 \times 3}$

= $[\begin{matrix} ω^{2} & ω^{3} & ω^{4} \\ ω^{3} & ω^{4} & ω^{5} \\ ω^{4} & ω^{5} & ω^{6} \end{matrix}]$ = $[\begin{matrix} ω^{2} & 1 & ω \\ 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \end{matrix}]$

$p^{2}$ = $[\begin{matrix} ω^{2} & 1 & ω \\ 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \end{matrix}]$ $[\begin{matrix} ω^{2} & 1 & ω \\ 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \end{matrix}]$ = $[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ =0

$∴ P^{2} = 0$ , when $n$ is a multiple of 3 .

$P^{2} \neq 0$ , when $n$ is not a multiple of 3 .

$\Rightarrow n = 57$ is not possible.

$∴ n = 55, 58, 56$ is possible.

Matrices and Determinants 1 Question 13

13. Let $ω$ be a complex cube root of unity with $ω \neq 0$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j}$ . Then, $P^{2} \neq 0$ when $n$ is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 1 Question 13

13. Let ω be a complex cube root of unity with ω≠0 and P=[pij] be an n×n matrix with pij=ωi+j. Then, P2≠0 when n is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

13. Let $ω$ be a complex cube root of unity with $ω \neq 0$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j}$ . Then, $P^{2} \neq 0$ when $n$ is equal to