Engineering

Mathematics

Introduction to Determinants

Question

$∣ \begin{array}{ccc} x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3} \end{array} ∣ = xyz (x - y) (y - z) (z - x)$

True

False

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Let $A = ∣ \begin{array}{ccc} x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3} \end{array} ∣$

Taking x, y, z common from C₁, C₂, C₃ respectively, we get,

A = xyz ∣ \begin{array}{ccc} 1 & 1 & 1 \\ x & y & z \\ x^{2} & y^{2} & z^{2} \end{array} ∣

Applying C₂ → C₂ – C₁ and C₃ → C₃ – C₁, we get,

A = xyz ∣ \begin{array}{ccc} 1 & 0 & 0 \\ x & y - x & z - x \\ x^{2} & y^{2} - x^{2} & z^{2} - x^{2} \end{array} ∣

Taking (y – x) and (z – x) common from C₂ and from C₃

A = xyz (y - x) (z - x) ∣ \begin{array}{ccc} 1 & 0 & 0 \\ x & 1 & 1 \\ x^{2} & y + x & z + x \end{array} ∣

Expanding along R₁,

A = xyz(y – x) (z – x) (z + x – y – x)

A = xyz(x – y) (y – z) (z – x)

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