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..
You need to be more specific with questions, especially ones that involve undetermined quantities.
Does the question say something like:
"Assume are distinct real numbers. Determine whether or not the matrix is invertible, or give necessary and sufficient conditions for the matrix to be invertible."
If this is what the question is asking, and if the determinant is indeed
then we know that
is invertible .