Let A, B be a pair of matrices with regular inertia. If HA + A* H and HB + B*H are both positive definite for some Hermitian matrix H then all matrices in conv(A, A(-1), B, B-1) have identical regular inertia. This, in turn, implies that both conv(A, B) and conv(A, B-1) consist of non-singular matrices. In general, neither of the converse implications holds. In this paper we seek situations where they do hold, in particular, when A and B are real 2 x 2 matrices. Several aspects of the above statements for n x n matrices are discussed. A connection to the characterization of the convex hull of matrices with regular inertia is introduced. Differences between the real and the complex case are indicated. (C) 2002 Elsevier Science Inc. All rights reserved.36083104