Let S be a regular semigroup. An inverse subsemigroup \(S^ 0\) of S is called an inverse transversal for S if \(S^ 0=S^ 0SS^ 0\) and each \(a\in S\) has a unique inverse \(a^ 0\in S^ 0\). We shall only speak about regular semigroups containing an inverse transversal. In a recent paper [ibid. 34, 459-474 (1983; Zbl 0537.20033)] the authors have given a structure theorem for such semigroups, now they develop a matrix type theory for them. Let \(S^ 0\) be an inverse transversal for S, and \(E=\{x\in S:\quad x=xx^ 0\},\quad F=\{x\in S:\quad x=x^ 0x\}.\) S is called relatively rigid with respect to S if \((\forall e,f,v\in E)\quad ((ev=fv\wedge e^ 0=f^ 0)\Rightarrow e=f,\) and dually for F. It is proved that if S can be imbedded in a Rees matrix semigroup over an inverse semigroup then S is relatively rigid with respect to \(S^ 0\). If S is relatively rigid with respect to \(S^ 0\) then S can be imbedded in a Rees matrix semigroup over the semigroup of order ideals of \(S^ 0\). There are more results. Some examples are provided to show that the assumptions under which the results have been obtained are ''best possible''.