  We consider solutions of the $2\times 2$ matrix Hamiltonian of physical
systems within the context of the asymptotic iteration method. Our technique is
based on transformation of the associated Hamiltonian in the form of the first
order coupled differential equations. We construct a general matrix Hamiltonian
which includes a wide class of physical models. The systematic study presented
here reproduces a number of earlier results in a natural way as well as leading
to new findings. Possible generalizations of the method are also suggested.
