Here's the problem:

Let $\displaystyle f$ be a function defined on the set of $\displaystyle n \times n$ matrices with entries from the field $\displaystyle F$ such that

$\displaystyle f(A + B) = f(A) + f(B)$,

$\displaystyle f(\lambda A) = \lambda f(A)$,

$\displaystyle f(AB) = f(BA)$.

Prove that there is an element $\displaystyle \alpha_0 \in F$ such that $\displaystyle f(A) = \alpha_0 \text{trace} (A)$.

I know that it can be shown that $\displaystyle f$ has the same value for all matrices in a similarity class, but I have no idea where to go from there. Any insight is appreciated!