Prove the following :

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Question

Prove the following : $∣ \begin{array}{ccc} y + z & z & y \\ z & z + x & x \\ y & x & x + y \end{array} ∣ = 4 xyz$

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Apply C₁ – (C₂ + C₃) to make one zero we get
$△ = ∣ \begin{array}{ccc} 0 & z & y \\ - 2 x & z + x & x \\ - 2 x & x & x + y \end{array} ∣$
Take out – 2x from C₁ and then apply R₂ – R₃
$Δ = - 2 x ∣ \begin{array}{ccc} 0 & z & y \\ 0 & z & - y \\ 1 & x & x + y \end{array} ∣$
= – 2x(– zy – zy) = 4 xyz.