Prove that : = (a – b)(b – c)(c

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

cubic and higher degree equation

Question

$∣ \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{3} & c^{3} \end{array} ∣$
Prove that : = (a – b)(b – c)(c – a)(a + b + c).

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Now,

∣ \begin{array}{ccc} 1 & a & a^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{array} ∣

[R'₂ = R₂ – R₁ and R'₃ = R₃ – R₁]

= ∣ \begin{array}{ccc} 1 & a & a^{3} \\ 0 & b - a & b^{3} - a^{3} \\ 0 & c - a & c^{3} - a^{3} \end{array} ∣

= (b - a) (c - a) ∣ \begin{array}{ccc} 1 & a & a^{3} \\ 0 & 1 & b^{2} + ab + a^{2} \\ 0 & 1 & c^{2} + ac + a^{2} \end{array} ∣

= – (b – a) (c – a) (b² – c² + a(b – c))

= – (b – a) (c – a) (b – c) (a + b + c)

= (a – b) (b – c) (c – a) (a + b + c)