Using properties of determinants prove that: |(1,1,1),(a,b,c),(bc,ca,ab)| = (a-b)(b-c)(c-a)

Question

Using properties of determinants prove that:

\(\begin{vmatrix} 1 &1 & 1 \\[0.3em] a& b & c \\[0.3em] bc & ca & ab \end{vmatrix}\) = (a-b)(b-c)(c-a).

Answer 1

\(\begin{vmatrix} 1 &1 & 1 \\[0.3em] a& b & c \\[0.3em] bc & ca & ab \end{vmatrix}\)

= \(\begin{vmatrix} 0&0 & 1 \\[0.3em] a-b& b-c & c \\[0.3em] bc - ca & ca -ab & ab \end{vmatrix}\)[C₁’ = C₁ - C₂ & C₂’ = C₂ - C₃]

= \(\begin{vmatrix} 0&0 & 1 \\[0.3em] a-b& b-c & c \\[0.3em] c(b - a ) & a(c -b) & ab \end{vmatrix}\)

= (a-b)(b-c) \(\begin{vmatrix} 0&0 & 1 \\[0.3em] a-b& b-c & c \\[0.3em] -c & -a& ab \end{vmatrix}\)[C₁’ = C₁/(a - b) & C₂’ = C₂/(b - c)]

= (a - b)(b - c)[0 + 0 + 1{ - a - ( - c)}] [expansion by first row]

= (a - b)(b - c)(c - a)