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We introduce a family of weight matrices W of the form T(t)T*(t), T(t) = e(At)eDt(2), where A is a certain nilpotent matrix and D is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size N x N and depend on N parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second-order differential equation with differential coefficients that are matrix polynomials F-2, F-1 and F-0 (independent of n) of degrees not bigger than 2, 1 and 0, respectively. For size 2 x 2, we find an explicit expression for a sequence of orthonormal polynomials with respect to W. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.