In this report, we investigate some problems of effectivity, related to algebraic residue theory. We show how matrix techniques based on Bezoutian formulations, enable us to derive new algorithms for these problems, as well as new bounds for the polynomials involved in these computations. More precisely, we focus on the computation of relations of algebraic dependencies between $n+1$ polynomials in $n$ variables and show how to deduce the residue of $n$ polynomials in $n$ variables. Applications for testing the properness of a polynomial map, for computing its Lojasiewicz exponent, and for inverting polynomial maps are also considered. We also show how Bezoutian matrices, enable us to compute a non-trivial multiple of the resultant on any irreducible algebraic variety and decompose an algebraic variety into irreducible components.