The determinant of this matrix

| a+b b+c c+a |
| sqrt(ab) sqrt(bc) sqrt(ca) |
| ab/(a+b) bc/(b+c) ca/(c+a) |

The task is to calculate this determinant, writing it as a product ..

Problem is, if I just go at it using Sarrus Rule, I just end up with a bunch of terms and I have no real clue what they have in common and how exactly to tie them up ..

I end up with a sum consisting of:

[ (a+c)*sqrt(bc) + (b+c)*sqrt(ac) ] * ab/(a+b) ..

And its two analogues .. It's probably just some easy algebra involving the grouping of certain terms, but I've been at it for a while now and I'm just not figuring it out ..

Help would be appreciated :)

EDIT: The reason I posted this here is because the solution to this most likely doesn't involve any elements from Matrix calculation .. Just arranging the terms you get in some order, grouping and factoring to get a product of paranthesis ..

Hey I tryied playing with it,

here is what I ended with

(abc)^(1/2)*[c*(a+b)*[a^[1/2]/(a+c)-b^[1/2]/(c+b)]- a*(b+c)*[b^[1/2]/(a+b)-c^[1/2]/(c+a)]+ b*(c+a)*[c^[1/2]/(c+b)-a^[1/2]/(b+a)]]

I couldn't make it look all good in latex, but it look great on matlab or wolfram =)


(abc)^(1/2)*[c*(a+b)*[a^[1/2]/(a+c)-b^[1/2]/(c+b)]- a*(b+c)*[b^[1/2]/(a+b&

I tryied some rule I know with determinant, nothing helped me.

Hope it helped