We study those exactness properties of a regular category C that can be expressed in the following form: for any diagram of a given ‘finite shape’ in C, a given canonical morphism between finite limits built from this diagram is a regular epimorphism. The main goal of the paper is to characterize essentially algebraic categories which satisfy this property via (essential versions of) linear Mal'tsev conditions, which are known to correspond to the so-called matrix properties. We then apply this characterization, along with our earlier work on preservation of exactness properties by pro-completions, to prove that these exactness properties can be reduced to matrix properties already in the general setting of regular categories.