In this short paper, we characterize the upper bound epsilon* for the parasitic parameter epsilon in a singularly perturbed systems, which ensures stability of such a system if 0 <epsilon <epsilon*. At the same time, a method is established to testify the system stability without the slow-fast decomposition required in the existing literature. It will be shown that this upper bound is just the minimum positive eigenvalue of a matrix pair, which is explicitly constructed from the system matrix. This reveals a direct relationship between the stability bound and the system matrix and may be useful in the study of robust-control problems for such systems. (C) 1997 Elsevier Science Inc.