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We consider solutions of the $2\times 2$ matrix Hamiltonian of physicalsystems within the context of the asymptotic iteration method. Our technique isbased on transformation of the associated Hamiltonian in the form of the firstorder coupled differential equations. We construct a general matrix Hamiltonianwhich includes a wide class of physical models. The systematic study presentedhere reproduces a number of earlier results in a natural way as well as leadingto new findings. Possible generalizations of the method are also suggested.

Solution of spin-boson systems in one and two-dimensional geometry via the asymptotic iteration method