I am to prove that adjoint of(AB)= adjoint of B times adjoint of A

u_ik=sum(j)[(a_ij)*(b_jk)]
I know then,I may take tarnspose of both sides so that we have:[(u_ki)~]=sum(j){[(b_kj)~][(a_ji)~]}
then [U~]=[B~][A~]
Then,we are done.But,this proves for real elements.I am not sure that this proves also for imaginary elements...
Please help.

To prove it for the complex case you have only to note that, for any z_1,z_2\in\mathbb{C}, \overline{z_1}\overline{z_2}=\overline{z_1z_2}.

That 's a different problem.How to show the proof is still valid if we are dealing with the complex matrices?

You need to use the clue that I gave you. When you write down B^{\dagger}A^{\dagger} you are going to have products of complex numbers of the form \overline{z_1}\overline{z_2}, which you must rewrite as \overline{z_1z_2}.