In this thesis, we introduce and study different definitions of positive matrix; entry-wise positive,positive definite, conditionally positive and totally positive. An m -by- n matrix A is calledtotally nonnegative (positive), TN ( TP ), if all minors of A are nonnegative (positive). We write A 2 TN ( TP ). if only the j -by - j minors, where 1 _ j _ k , are nonnegative (positive), we write TNk ( TPk ). The special in which the matrices are square, in TN 2 and all 2-by-2 principalminors based upon consecutive indices are positive, is denoted by TN + 2 . If A = [ aij ] is an m -by- n entery-wise nonnegative matrices, then A ( t ) = [ at ij ] denotes the Hadamard power of A for any positive real number t . If A;B 2 TN 2 ( TP 2), then the Hadamard product A _ B is a and A ( t ) for every t _ 0 are TN 2 ( TP 2) matrices. However, TNk ( TPk ) is not closed under Hadamard multiplication when k _ 3. We say that entry-wise nonnegative (positive) matrix A is eventually TN (TP), if there exists a T _ 0 such that A ( t ) is TN (TP) for all t _ T . In this thesis, also, positive Hadamard powers of entery-wise positive (nonnegative) matrices, those that are eventually totally positiv, totally nonnegative and doubly positive are charactrized, For example, for matrices with at least four rows and columns, Hadamard powers greater than oneof TP matrices need not be totally positive, but they are eventually positive. The first two chapters of this thesis provide the preliminaries for discusion of positive matrices.Such matrial can be found in any linear algebra text book. In chapters three and four, totally positive matrices are studied. Main resuts are presented in chapter five.