Let `omega` be a complex cube root of unity with `omega!=1a n dP=[p_(i j)]` be a `nxxn` matrix withe `p_(i j)=omega^(i+j)dot` Then `p^2!=O ,w h e nn=` a.`57` b. `55` c. `58` d. `56`

Options

A. 57
B. 55
C. 58
D. 56

Answer

B, C, D

Explanation

`P=[(omega^(2),omega^(3),omega^(4),...,omega^(n+1)),(omega^(3),omega^(4),omega^(5),...,omega^(n+2)),(vdots,vdots,vdots,vdots,vdots),(omega^(n+1),omega^(n+2),...,...,omega^(2n))]`
or
`P^(2)=[(omega^(4)+omega^(6)...,omega^(5)+omega^(7)+...,...,...),(omega^(5)+omega^(7)+...,...,...,...),(vdots,vdots,vdots,...),(omega^(n+2)+omega^(n+4)...,...,...,omega^(2n+2)+omega^(2n+4)...)]`
= Null matrix if n is a multiple of 3.