In this article we consider experimental settings in which it is desired to optimally comparevtest treatments to a standard or control treatment and the experimental units are to be arranged inbblocks of sizek. This problem has received a good deal of attention in recent years. Majumdar and Notz (1983) developed some sufficient conditions that can be used to establish theA-optimality of balanced treatment block designs (BTBD's) in these situations, and several authors have since used these sufficient conditions to show theA-optimality of some specific BTBD's as well as characterize some infinite families ofA-optimal BTBD's. Here we develop some further sufficient conditions forA-optimality that generalize those given by Majumdar and Notz (1983) and can often be used to establish theA-optimality of BTBD's not covered by the results of Majumdar and Notz (1983), as well as theA-optimality of certain types of designs called group-divisible treatment designs. Several examples are given to illustrate how the sufficient conditions obtained here can be applied.