Question

Prove that `|(1/a,bc,b+c),(1/b,ca,c+a),(1/c,ab,a+b)|` = 0

Answer 1

`|(1/a,bc,b+c),(1/b,ca,c+a),(1/c,ab,a+b)|`

= `1/"abc" |(a/a,abc,a(b+c)),(b/b,abc,b(c+a)),(c/c,abc,c(a+b))|`   ....(Multiply R₁ by a, R₂ by b, R₃ by c and divide the determinant by abc)

= `"abc"/"abc" |(1,1,a(b + c)),(1,1,b(c+a)),(1,1,c(a + b))|`    ....(Take out abc from C₂ = 0 [∵ C₁ ≡ C₂])

Hence proved.