If ω is one of the imaginary cube roots of unity, find the value of .

Engineering

Mathematics

Introduction to Determinants

Question

If ω is one of the imaginary cube roots of unity, find the value of $∣ \begin{array}{ccc} 1 & ω^{3} & ω^{2} \\ ω^{3} & 1 & ω \\ ω^{2} & ω & 1 \end{array} ∣$ .

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$∆ = ∣ \begin{array}{ccc} 1 & ω^{3} & ω^{2} \\ ω^{3} & 1 & ω \\ ω^{2} & ω & 1 \end{array} ∣$

we know ω³ = 1

\Rightarrow Δ = ∣ \begin{array}{ccc} 1 & 1 & ω^{2} \\ 1 & 1 & ω \\ ω^{2} & ω & 1 \end{array} ∣

R₁ = R₁ – R₂

\Rightarrow Δ = ∣ \begin{array}{ccc} 0 & 0 & ω^{2} - ω \\ 1 & 1 & ω \\ ω^{2} & ω & 1 \end{array} ∣

Expanding along R₁

⇒ Δ = 0 + 0 + (ω² – ω) (ω – ω²)

⇒ Δ = – (ω² – ω)²

⇒ Δ = – (ω⁴ + ω² – 2ω³)

⇒ Δ = – (ω + ω² – 2)

As 1 + ω + ω² = 0

⇒ Δ = – (– 1 – 2)

⇒ Δ = 3

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