We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].