Chapter 2 - Simultaneous linear algebraic equations

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The mathematical analysis of linear physicochemical systems often results in models consisting of sets of linear algebraic equations. In addition, methods of solution of nonlinear systems and differential equations use the technique of linearization of the models, thus requiring the repetitive solution of sets of linear algebraic equations. These problems may range in complexity from a set of two simultaneous linear algebraic equations to a set involving 1000 or even 10,000 equations. The solution of a set of two to three linear algebraic equations can be obtained easily by the algebraic elimination of variables or by the application of Cramer’s rule. However, for systems involving five or more equations, the algebraic elimination method becomes too complex, and Cramer’s rule requires a rapidly escalating number of arithmetic operations, too large even for today’s high-speed digital computers. Various methods of solution for a set of linear algebraic equations is presented in this chapter. In developing systematic methods for the solution of linear algebraic equations and the evaluation of eigenvalues and eigenvectors of linear systems we will make extensive use of matrix-vector notation. For this reason, and for the benefit of the reader, a review of selected matrix and vector operations is also given.