The solution to a set of simultaneous equations is of the form A-1 B = X where A-1 is the inverse of A in the equation AX= B. The purpose of this study is to obtain an exact A-1 through the use of rational arithmetic, and to study the behavior of rational numbers when used in arithmetic calculations. This study describes a matrix inversion program written in SPS II, utilizing the concept of rational arithmetic. This program, using the Gaussian elimination matrix inversion method, is compared to the same method written in Fortran. Gaussian elimination was used by this study because of its simplicity and speed of inversion. The Adjoint method was ruled out because of its complexity and relative lack of speed when compared with Gaussian elimination. The Fortran program gives only an approximate inverse due to the rounding error while the rational arithmetic program gives an exact inverse.