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

I satrted this as U=adjoint of AB

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.