An r-matrix is a matrix with symbols in {0, 1, . . . , r − 1}. A matrix is simple if it has no repeated columns. Let the support of a matrix F , supp(F ) be the largest simple matrix such that every column in supp(F ) is in F . For a family of r-matrices F, we define forb(m, r, F) as the maximum number of columns of an m-rowed, r-matrix A such that F is not a row-column permutation of A for all F ∈ F. While many results exist for r = 2, there are fewer for larger numbers of symbols. We expand on the field of forbidding matrices with r-symbols, introducing a new construction for lower bounds of the growth of forb(m, r, F) (with respect to m) that is applicable to matrices that are either not simple or have a constant row. We also introduce a new upper bound restriction that helps with avoiding non-simple matrices, limited either by the asymptotic bounds of the support, or the size of the forbidden matrix, whichever is larger. Continuing the trend of upper bounds, we represent a well-known technique of standard induction as a graph, and use graph theory methods to obtain asymptotic upper bounds. With these techniques we solve multiple, previously unknown, asymptotic bounds for a variety of matrices. Finally, we end with block matrices, or matrices with only constant row, and give bounds for all possible cases.