Proof of 2 rank Matrix

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?

Delta2