Prove that

Engineering

Mathematics

Cube Roots of Unity

Introduction to Determinants

Properties of Determinant

Question

Prove that $|\begin{array}{ccc} 1 & ω & ω^{2} \\ ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array}| = 0$

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Solving given determinant, we get

[1(ω³ – 1) – ω(ω² – ω²) + ω²(ω – ω⁴)] ....... as (ω³ = 1)

⇒ [(1 – 1) – ω(0) + ω²(ω – ω)]

⇒ 0 + 0 + 0

⇒ 0

Hence proved