The object of this paper is to develop the ideas introduced in the author's paper [1] on matrices which generate families of polynomials and associated infinite series. A family of infinite one-subdiagonal non-commuting matrices Qm is defined, and a number of identities among its members are given. The matrix Q1 is applied to solve a problem concerning the derivative of a family of polynomials, and it is shown that the solution is remarkably similar to a conventional solution employing a scalar generating function. Two sets of infinite triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al. [24-26].