  Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials with
respect to the sesquilinear form $\langle P(x),Q(x)\rangle_W=\int P(x)
W(x)\operatorname{d}\mu(x)(Q(x))^{\top}$, $P(x),Q(x)\in\mathbb R^{p\times
p}[x]$, 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 bi-orthogonal polynomials with respect to $\langle
\cdot,\cdot\rangle_W$ 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 Dur\'an et al are revisited. Finally, we extend these results to
the non-Abelian 2D Toda lattice hierarchy.
