  In this work the interplay between matrix biorthogonal polynomials with
respect to a matrix of linear functionals, the $k$-th associated matrix
polynomials and the second kind matrix functions, is studied in terms of
quasideterminants. A sort of Poincar\'{e}'s theorem for the ratio of two
consecutive matrix functions solutions of a linear difference equation is also
presented. Some new formulas connecting these families of matrix functions are
given.
