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Differentiability in an Interval

Let ω be a ...

Question

Let $ω$ be a complex cube root of unity with $ω \neq 1$ and $P = [p_{i j}]$ be a n $\times$ n matrix with $p_{i j} = ω^{i + j}$ . Then $P^{2} \neq 0$ , when $n =$

A

57

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B

55

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C

58

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D

56

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Solution

The correct options are
A $55$
C $58$
D $56$
$P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ω^{2} & ω^{3} & ω^{4} & . . . . & ω^{n + 1} ω^{3} & ω^{4} & ω^{5} & . . . . & ω^{n + 2} ω^{4} & ω^{5} & ω^{6} & . . . . & ω^{n + 3} . . . & . . . & . . . & . . . . & . . . ω^{n + 1} & ω^{n + 2} & . . . & . . . & ω^{2 n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$P = ω^{2} ω^{3} ω^{4} . . . . . ω^{n + 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} . . . & . . . & . . . & . . . . & . . . 1 & ω & ω^{2} & . . . & ω^{n - 1} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$P = ω^{\frac{n (n + 3)}{2}} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} . . . & . . . & . . . & . . . . & . . . 1 & ω & ω^{2} & . . . & ω^{n - 1} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
Now, $P^{2} = ω^{n^{2} + 3 n} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} . . . & . . . & . . . & . . . . & . . . 1 & ω & ω^{2} & . . . & ω^{n - 1} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} 1 & ω & ω^{2} & . . . . & ω^{n - 1} . . . & . . . & . . . & . . . . & . . . 1 & ω & ω^{2} & . . . & ω^{n - 1} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
Since, $P^{2} \neq O$
$\Rightarrow 1 + ω + ω^{2} + . . . . ω^{n - 1} \neq 0$
$\Rightarrow \frac{1 - ω^{n}}{1 - ω} \neq 0$
If n is a multiple of 3, then $1 - ω^{n} = 0$
So, n must not be a multiple of 3.
Hence, n cannot be 57.

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