Let $V$ be the vector space consists of all $n\times n$ real matrices, and $f$ is a nonvanishing linear function on $V$ such that $$f(AB)=f(BA),\ \forall\ A,B\in V.$$ Show that $g(A,B)=f(AB)$ is a non-degenerate bilinear function on $V$.
What I could show is only "g" is bilinear.