A new supersymmetric approach to dynamical symmetries for matrix quantum systems is explored. In contrast to standard one-dimensional quantum mechanics where there is no role for an additional symmetry due to nondegeneracy, matrix Hamiltonians allow nontrivial residual symmetries. This approach is based on a generalization of the intertwining relations familiar in SUSY quantum mechanics. The corresponding matrix supercharges, of first or of second order in derivatives, lead to an algebra which incorporates an additional block diagonal differential matrix operator (referred to as a 'hidden' symmetry operator) found to commute with the super-Hamiltonian. We discuss some physical interpretations of such dynamical systems in terms of spin 1/2 particle in a magnetic field or in terms of coupled channel problem. Particular attention is paid to the case of transparent matrix potentials.