Let V be the set of all complex square matrices. For $\displaystyle D \in V$ define $\displaystyle T_D : V \rightarrow V$ by $\displaystyle T_D(A) = DA$.

Using the inner product $\displaystyle <X,Y>=tr(XY^*)$, find the adjoint of $\displaystyle T_D$.

I feel I'm missing something because I can't arrive at an answer.