....(where ω is the cube root of unity)

Engineering

Mathematics

Introduction to Determinants

Question

$∣ \begin{array}{lll} 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array} ∣ =$ ....(where ω is the cube root of unity)

– 1

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⇒ ω is cube root of unity
⇒ ω³ – 1 = 0

By operation of matrix, (5),

∣ \begin{array}{lll} 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array} ∣

= 1(ω³ – 1) – ω(ω² – ω²) + ω²(ω – ω⁴) = 2ω³ – 1 – ω⁶ = 2 – 1 – 1 = 0

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