Let - be a complex cube root of unity with - 1 and P-pij- be a n- n matrix with pij- i-j- Then P2- 0- when n-

P=[P_ij]_n× n P^2≠ 0 P_ij=ω ^i+j P= [ ω ^2 amp; 1 amp; ω nbsp; amp; ω ^2 amp; 1 amp; ... nbsp; 1 amp; ω nbsp; amp; ω ^2 amp; 1 ...

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Standard X

Mathematics

Null Matrix

Let ω be a co...

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 $n =$

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Solution

The correct option is D 55
$P = [P_{i j}]_{n \times n} P^{2} \neq 0$

$P_{i j} = ω^{i + j}$

$P = {⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ω^{2} & 1 & ω & ω^{2} & 1 & . . . 1 & ω & ω^{2} & 1 & ω & . . . ω & ω^{2} & 1 & ω & ω^{2} & . . . . . . & . . . & . . . & . . . & . . . & . . . \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}_{n \times n}$

$P^{2} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ω^{2} & 1 & ω & ω^{2} & 1 & . . . 1 & ω & ω^{2} & 1 & ω & . . . ω & ω^{2} & 1 & ω & ω^{2} & . . . . . . & . . . & . . . & . . . & . . . & . . . \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ω^{2} & 1 & ω & ω^{2} & 1 & . . . 1 & ω & ω^{2} & 1 & ω & . . . ω & ω^{2} & 1 & ω & ω^{2} & . . . . . . & . . . & . . . & . . . & . . . & . . . \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$(ω^{4} + 1 + ω^{2}) + (ω^{4} + 1 + ω^{2}) + . . . = 0$

Only when n is a multiple of 3.

$∴ n$ can be $55, 58, 56 (p^{2} \neq 0)$

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Let ω be a complex cube root of unity with ω≠1 and P=[pij] be a n×n matrix with pij=ωi+j. Then P2≠0; when n=

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 $n =$