Proving a Matrix Inner Product

Question

I am given a matrix inner product on square matrices defined as $\langle A,B\rangle=tr(AB^t)$, where $M^t$ denotes the transpose. I am asked to prove that this is indeed an inner product. We go by 3 definitions for inner product:
$\langle A+B,C\rangle=\langle A,C\rangle+\langle B,C\rangle$
$\langle A,B\rangle=\overline{\langle B,A\rangle}$
$\langle A,A\rangle\geq0$, in particular, $\langle A,A\rangle=0\iff A=0$
I have proven that the defined inner product fits the first and the last definition, but I am having trouble going through the conjugate symmetry proof. This is what I have so far:
$\overline{\langle B,A\rangle}=\overline{tr(BA^t)}=tr(\overline{BA^t})=tr((\overline{A})\overline{(B^t)})^t$

Answer 1

Hint: for any compatible matrices $A,B$, we have $$ \operatorname{trace}(AB) = \operatorname{trace}(BA) $$