Question

Using properties of determinant prove that :

|(1,1,1),(a,b,c),(bc,ca,ab)| = (a - b)(b - c)(c - a)

Answer 1

Lat = |(1,1,1),(a,b,c),(bc,ca,ab)|

= |(1,0,0),(a,b - a,c - b),(bc,ca - bc,ab - ca)|

[By C₂ → C₂ - C₁ and C₃ → C₃ - C₂]

= |(1,0,0),(a,-(a - b),-(b - c),(bc,c(a - b),a(b - c)|

= (a - b)(b - c)|(1,0,0),(a,-1,-1),(bc,c,a)|

[Taking common (a - b) from C₂ & (b - c) from C₃

= (a - b)(b - c).1(-a + c)

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