Let ω be a complex cube root of unity with ω ≠ 1 and P = p ij be a n x n matrix with p ij =…

Let  ω  be a complex cube root of unity with ω ≠ 1  and P = p ij  be a n x n matrix with p ij = ω i + j . Then P 2 ≠ 0 , when n =
  1. 57
  2. 55
  3. 58
  4. 56

Solution

n   = 1                                 n = 2
P = ω 2                            P = ω 2 ω 3 ω 3 ω 4 = ω 2 1 1 ω
P 2 = ω 4 ≠ 0                    P 2 = ω 4 + 1 ....... ....... ....... ≠ 0
n = 3
P = ω 2 1 ω 1 ω ω 2 ω ω 2 1   ω 2 1 ω 1 ω ω 2 ω ω 2 1 = 0 0 0 0 0 0 0 0 0
Similarly P 2 ≠ 0   when n is not multiple of 3.

Asked in: JEE Advanced 2013 (Paper 2)

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