The correct option is D 55
P=[Pij]n×n P2≠0
Pij=ωi+j
P=⎡⎢
⎢
⎢
⎢⎣ω21ωω21...1ωω21ω...ωω21ωω2.....................⎤⎥
⎥
⎥
⎥⎦n×n
P2=⎡⎢
⎢
⎢
⎢⎣ω21ωω21...1ωω21ω...ωω21ωω2.....................⎤⎥
⎥
⎥
⎥⎦⎡⎢
⎢
⎢
⎢⎣ω21ωω21...1ωω21ω...ωω21ωω2.....................⎤⎥
⎥
⎥
⎥⎦
(ω4+1+ω2)+(ω4+1+ω2)+...=0
Only when n is a multiple of 3.
∴n can be 55,58,56 (p2≠0)