Matrices and Determinants 1 Question 13

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

(2013 Adv.)

(a) 57

(b) 55

(c) 58

(d) 56

Show Answer

Answer:

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

Solution:

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

with

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

$\therefore$ When $n=1$

$ \begin{array}{rlrl} P & =\left[p_{i j}\right]_{1 \times 1}=\left[\omega^{2}\right] \\ \Rightarrow P^{2} & =\left[\omega^{4}\right] \neq 0 \end{array} $

$\therefore$ When $n=2$

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

=$\begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} $= $\begin{bmatrix} \omega^{2} & \omega^{3} \\ \omega^{3} & \omega^{4} \end{bmatrix} $= $\begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega \end{bmatrix} $

$ P^{2}= \begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega & \end{bmatrix} $ $\begin{bmatrix} & \omega^{2} & 1 \\ & 1 &\omega \end{bmatrix} $

$ \Rightarrow \quad P^{2}=$ $\begin{bmatrix} \omega^{4}+1 & \omega^{2}+\omega \\ \omega^{2}+\omega & 1+\omega^{2} \end{bmatrix}$ $\neq 0$

\omega^{2}

\omega

When $n=3$

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

=$\begin{bmatrix} \omega^{2} & \omega^{3} & \omega^{4} \\ \omega^{3} & \omega^{4} & \omega^{5} \\ \omega^{4} & \omega^{5} & \omega^{6} \end{bmatrix}$ =$\begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix}$

$p^2$=$\begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix}$ $\begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix}$ =$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$=0

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

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

$\Rightarrow \quad n=57$ is not possible.

$\therefore \quad n=55,58,56$ is possible.



NCERT Chapter Video Solution

Dual Pane