Invertibility of 3x3 matrix.

there is a 3 X 3 matrix ,

A = [ a b c]

[ b c a ]

[ c a b]

where a , b c are all distinct... can you prove that a + b + c is not 0 ?

the actual question is to prove whether it is invertible. If it is invertible, then its determiniant should not be 0 right? so I got |A| as -(a^3 + b^3 + c^3 ) + 3abc .. which is equal to -(a+b+c)(a^2 + b^2 + c^2 - ab - bc -ca ) which can be factored to

-1/sqrt(2) * (a+b+c)((a-b)^2 + (b-c)^2 + (c-a)^2)

the second part cant be zero as a, b, c are distinct.... so I have to prove that a + b + c is also not 0..