We study a number of characteristics of the inverses of the elements of a family of Pascal-like matrices that are defined by Riordan arrays. We give several forms of the bivariate generating function of these inverses, along with four different closed-form expressions for the general element of the inverse. We study the row sums and the diagonal sums of the inverses, and the first column sequence. We exhibit the elements of the first column sequence of the inverse matrix as the moments of a family of orthogonal polynomials, whose coefficient array is again a Riordan array. We also give the Hankel transform of these latter sequences. Other related sequences are also studied.