Systems of Linear Equations and Matrices

This chapter introduces students to the basics of linear equations and their solutions using matrix algebra. Students practice converting systems to matrices and vice versa, then use Gauss-Jordan elimination to solve those systems. Systems are classified according to the nature of their solutions: none (inconsistent), unique, or infinitely many. The possible outcomes for 3x3 matrices are listed and used to illustrate the general processes, along with the general form for reduced echelon form of any matrix. A few elementary applications are given, including population migration, networks, general stochastic matrices, and contact matrices for epidemics. Homogeneous systems are introduced along with their relationship to inhomogeneous systems and their solutions. Parameterized solutions are introduced, and the process of obtaining them may be practiced in the associated software. The important topic of elementary matrices is introduced, and basic properties of these matrices are explored. Elementary matrices are used to justify the process for inverting matrices from the prior chapter, and it is shown that a matrix is invertible if and only if it is the product of elementary matrices.