Question about the trace map

Question

Let k be a field and $f:M_n(k)\to k$ be linear transformation such that $f(AB)=f(BA)$ for all $A,B \in M_n(k)$. Show that f is a scalar multiple of trace map.

I thought that it might be the determinant one but I'm not sure whether it is or not. I somehow have the intuition to solve the problem. I am searching for the guidance for this. Thanks in advance!

Answer 1

Hints. Denote by $E_{ij}$ the matrix whose only nonzero entry is a $1$ at the $(i,j)$-th position. For any pair of matrices $A$ and $B$, denote also by $[A,B]$ the commutator $AB-BA$.

1. Using the fact that $[E_{11},E_{12}]=E_{12}$ and $[E_{12},E_{21}]=E_{11}-E_{22}$, prove that the linear span of all commutators is the subspace of all traceless matrices.
2. Hence, by writing $M$ as $\left(M-\operatorname{tr}(M)E_{11}\right)+\operatorname{tr}(M)E_{11}$, prove that $f(M)=\operatorname{tr}(M)f(E_{11})$.