The method of Conjugate Gradients is known to converge for symmetric positive definite systems of equations. This paper applies it to non-symmetric and ill-conditioned matrices. In order to facilitate convergence, an approximate inverse is used to precondition the Conjugate Gradient method. This is achieved by applying Newton's method. Three versions of Newton's method are introduced to compute the approximate inverse. Convergence of each version is compared. Numerical experimentation is done for some known "ill-conditioned" problems.