Generalized Pascal matrices generate classes closed under multiplication

Classes of matrices which are the Hadamard product of a fixed lower triangular generating matrix P and any Toeplitz matrix are studied. These classes are generalizations of the special case when a Toeplitz matrix is generated by the vector of powers and P is either also Toeplitz or a Pascal triangle. The matrix P which lead to the class being closed under matrix multiplication is fully characterized. Explicit formulae for inverses are derived and commutativity of products within each class proven. Examples using lower triangular matrices with binomial coefficients are given.