If p -3 k and 1- - -2 are the cube roots of unity- then Delta - -egin vmatrix 1 - lomega - n - lomega -2 n lend vmatrix has the valueA- -2B- 0C- -D- 1

The correct option is B 0Applying C_1 → C_1 + C_2 + C_3, we get Δ = 1 + ω^n + ω^2n amp; ω^n amp; ω^2n 1 + ω^n + ω^2n amp; 1 amp; ω^n 1 + ω^n ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

If p '=3 k an...

Question

If $n \neq 3 k$ and $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & ω^{n} & ω^{2 n} ω^{2 n} & 1 & ω^{n} ω^{n} & ω^{2 n} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ has the value

0

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ω

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ω^{2}

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1

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Solution

The correct option is A $0$
Applying $C_{1} \to C_{1} + C_{2} + C_{3},$ we get
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 + ω^{n} + ω^{2 n} & ω^{n} & ω^{2 n} 1 + ω^{n} + ω^{2 n} & 1 & ω^{n} 1 + ω^{n} + ω^{2 n} & ω^{2 n} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & ω^{n} & ω^{2 n} 0 & 1 & ω^{n} 0 & ω^{2 n} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0,$
$(∵ 1 + ω^{n} + ω^{2 n} = 0,$ if n is not multiple of 3).

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Q. If

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Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 ω^{2 n} ω^{n} \end{matrix} \begin{matrix} ω^{n} 1 ω^{2 n} \end{matrix} \begin{matrix} ω^{2 n} ω^{n} 1 \end{matrix} ∣ ∣ ∣ ∣

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If n≠3k and 1,ω,ω2​ are the cube roots of unity, then Δ=∣∣ ∣ ∣∣1ωnω2nω2n1ωnωnω2n1∣∣ ∣ ∣∣ has the value

The correct option is A 0Applying C1→C1+C2+C3, we get Δ=∣∣ ∣ ∣∣1+ωn+ω2nωnω2n1+ωn+ω2n1ωn1+ωn+ω2nω2n1∣∣ ∣ ∣∣=∣∣ ∣ ∣∣0ωnω2n01ωn0ω2n1∣∣ ∣ ∣∣=0, (∵1+ωn+ω2n=0,if n is not multiple of 3).

If $n \neq 3 k$ and $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & ω^{n} & ω^{2 n} ω^{2 n} & 1 & ω^{n} ω^{n} & ω^{2 n} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ has the value