A posteriori inference is one of three kinds of inference that underlie the processing of knowledge patterns with probabilistic uncertainty in intelligent decision making systems using alge braic Bayesian networks (ABNs). In this paper, the key terms and formulations of theorems describing local a posteriori inference in algebraic Bayesian networks are given in a matrix vector language. The main result is that matrix equations are constructed for normalizing factors appearing in the formulas of a posteriori probabilities of proposition quanta and ideals of conjuncts. The matrix equations of local a priori inference formulated in general not only simplify the preparation of specifications of appropriate inference algorithms and make their implementation more transparent, but also open a possibility for application of classical mathematical techniques to the analysis of the properties of inference results.