Proving A Must Be of Rank 2: The 2x2 Matrix Dilemma

[LukasMont]

Homework Statement

4.1 Show that one may express any second rank matrix as the sum of a symmetric
and an antisymmetric matrix.

Relevant Equations

I was able to proof that any matrix could be constructed by adding a symmetric and antisymmetric matrix:

A= A/2 + A/2 + A'/2 - A'/2,
A= (A/2 + A'/2) + (A/2 - A'/2), where A' is the transposed matrix. Now,

A/2 + A'/2 is symmetric, since (A/2 +A'/2)' = A'/2 + A/2 (equal) and
A/2 - A'/2 is antisymmetric, since (A/2 - A'/2)' = - A'/2 + A/2= -(A/2 - A'/2).

My trouble is being to show A must be of rank 2. Any ideas?

 

[andrewkirk]

You don't have to show that A has rank 2. You are given that as a premise.
I don't know why they give you that as a premise though, because the theorem is true for any matrix, not just any rank-2 matrix, as your proof shows.
Anyway, you have proven what they asked you to.