his M.Sc. thesis is based on the following paper : R. Bhatia, J. Holbrook Riemannian geometry and matrix geometric means, Linear Algebra Appl. ??? (????), ???-???. Operator theorists, physicists, engineers and statisticians have long been interested in various averaging operations (means) on positive definite matrices. When just two matrices are involved the theory is very well developed. See the foundational work of Kubo and Ando [??]. The concept of geometric mean of two positive definite matrices was first introduced by Pusz and Woronowicz [??]. The studies of these mathematicians can not be easily extended to three matrices and it has been along-standing problem to define a natural geometric mean of three positive definite matrices. In some recent paper a new concept of geometric mean of two positive definite matrices has been achieved by identi- fying the geometric mean of A and B as the midpoint of geodesic joining A and B which is stated below. The basic definitions for this concept have been a lot of depth and ideas related to our work iired by references [?] and [??]. The thesis proceeds as follows : In the first chapter, we introduce the basic concepts and definitions needed.