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Given a matrix polynomial W(x), matrix bi-orthogonal polynomials with respect to the sesquilinear form < P(x), Q(x)>(W) = integral P(x)W(x) d mu(x)(Q(x))(inverted perpendicular), P, Q is an element of R-pxp [x], where mu(x) is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix biorthogonal polynomials with respect to <.,.>(W) and matrix polynomials orthogonal with respect to mu(x) are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial W(x) we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of W(x). For perturbations with a singular leading coefficient, several examples by Duran and coworkers are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.