Prove that a bilinear form is nondegenerate

Question

Let $n$ be a positive integer, $V=\mathbb{C}^{n\times n}$ and $$f(A,B)=n\mathrm{Tr}(AB) - \mathrm{Tr}(A)\mathrm{Tr}(B).$$ Let $$W=\{ A\in V\, |\, \mathrm{Tr}(A)=0 \} $$ and let $f_1=f|_{W}$.

Prove that $f_1$ is nondegenerate.

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What I have done:
We have that $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0 $, then $f_1(A,B)=n\mathrm{Tr}(AB)$. We have to prove that: $n\mathrm{Tr}(A,B)=0 \; \forall \beta \in W $ implies $A=0$. But I don't know where to go from there.

Answer 1

Hint: What is $f_1(A, ^t\overline{A})$?