It is obvious that between any two rows (columns) of an \(m\)-by-\(n\) totally nonnegative matrix a new row (column) may be inserted to form an \((m+ 1)\)-by-\(n\) \((m\)-by-\((n+1))\) totally nonnegative matrix. The analogous question, in which ``totally negative'' is replaced by ``totally positive'' arises, for example, in completion problems and in extension of collocation matrices, and its answer is not obvious. Here, the totally positive case is answered affirmatively, and in the process an analysis of totally positive linear systems, that may be of independent interest, is used.