Three methods for solving systems of linear equations: Comparing the advantages and disadvantages

Abstract

This paper aims to introduce some prevalent techniques which have been used to solve linear systems. Firstly, we introduce the simplest method: how to eliminate variables in which the original systems would not be changed. We then introduce Gaussian eliminations, which work on the augmented matrix derived from a linear system. To the end, we present Cramer’s rule, which computes the solution to a linear system based on matrix determinants. These methods have their application scenarios. For instance, eliminating variables has no limited condition for its operations. However, Cramer’s rule needs to be under the condition of square matrices. At the same time, Gaussian elimination requires three elementary row operations, and Gaussian elimination paves the way for computing the rank of matrices. The choice between these methods depends on coefficient matrices in terms of dimensions and matrix properties such as singularity.