  Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to
exactness properties in Category Theory. Some can be stated in the finitely
complete context while, in general, they can only be stated for regular
categories. We study when such a matrix condition implies the Mal'tsev
property. Our main results assert that, for both types of matrices, this
implication is equivalent to the corresponding implication restricted to the
context of varieties of universal algebras.
