Let $A,B \in \mathbb{C}^{n\times n}$. Define $ \langle A,B \rangle := \operatorname{Tr}(A^*B)$. Show that $\langle \cdot,\cdot \rangle $ is a scalar product on $\mathbb{C}^{n\times n}$.
I already proved the most properties. But I have trouble proving the homogeneity property.
Here is my solution:
$$\langle \lambda A,C \rangle = \operatorname{Tr}((\lambda A)^*C) =\operatorname{Tr}(\overline\lambda A^*C)=\overline\lambda \operatorname{Tr}(A^*C)=\overline\lambda \langle A,C\rangle.$$
But this is not the homogeneous property of complex scalar products, which is $ \langle \alpha x,y \rangle = \alpha \langle x,y \rangle$.
Can somebody tell me what did I do wrong?
P.S. The problem is from an old examine, so it can't be wrong.
This is just a difference of convention. The question assumes you want the second argument to be linear. (To change which argument is linear, just conjugate the inner product.) That's what physicists usually assume for quantum mechanics.