The framework of the well-known simplex algorithm for linear programming is used to analyze systems of linear algebraic equations and linear algebraic inequalities.

A sequence of basis matrices, where each matrix results from one row change from its predecessor, is used. The results developed are contained in two theorems: the first provides update formulas for the set of values that change when the basis matrix is updated, and the second characterizes the solution to the inequality problem using a cone defined in terms of columns of the basis matrix inverse.

The algebraic development is well presented; however, the Gaussian elimination mechanics lack the stability properties that are expected in a strong floating-point algorithm. For example, the proposed determination of rank is unreliable compared with the preferable orthogonal transformation-based singular value decomposition. The use of the basis matrix inverse has descriptive value in the statement of the second theorem, but in an implementation, the avoidance of explicit inverse calculations and the use of orthogonal matrix factorizations are numerically superior.

Experimentally, the implementation is shown to work well on a set of complete rank matrices varying in size up to 1,500-by-1,500, and with random elements uniformly distributed in (0,1); however, the testing of more extreme cases is needed in order to illustrate implementation limitations.

The paper ends with some suggested future work, including the need for further stability considerations.