  Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices
and the extremal sat-function for graphs, we introduce sat-type problems for
matrices. Let F be a family of k-row matrices. A matrix M is called
F-admissible if M contains no submatrix G\in F (as a row and column permutation
of G). A matrix M without repeated columns is F-saturated if M is F-admissible
but the addition of any column not present in M violates this property. In this
paper we consider the function sat(n,F) which is the minimum number of columns
of an F-saturated matrix with n rows. We establish the estimate
sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the
sat-function for a few small forbidden matrices.
