We will start this chapter off by looking at the application of matrices that almost every book on Linear Algebra starts off with, solving systems of linear equations.  Looking at systems of equations will allow us to start getting used to the notation and some of the basic manipulations of matrices that we’ll be using often throughout these notes.

Once we’ve looked at solving systems of linear equations we’ll move into the basic arithmetic of matrices and basic matrix properties.  We’ll also take a look at a couple of other ideas about matrices that have some nice applications to the solution to systems of equations.

One word of warning about this chapter, and in fact about this complete set of notes for that matter, we’ll start out in the first section doing a lot of the details in the problems, but towards the end of this chapter and into the remaining chapters we will leave many of the details to you to check.  We start off by doing lots of details to make sure you are comfortable working with matrices and the various operations involving them.  However, we will eventually assume that you’ve become comfortable with the details and can check them on your own.  At that point we will quit showing many of the details.

Here is a listing of the topics in this chapter.

Systems of Equations  In this section we’ll introduce most of the basic topics that we’ll need in order to solve systems of equations including augmented matrices and row operations.

Solving Systems of Equations  Here we will look at the Gaussian Elimination and Gauss-Jordan Method of solving systems of equations.

Matrices  We will introduce many of the basic ideas and properties involved in the study of matrices.

Matrix Arithmetic & Operations  In this section we’ll take a look at matrix addition, subtraction and multiplication.  We’ll also take a quick look at the transpose and trace of a matrix.

Properties of Matrix Arithmetic  We will take a more in depth look at many of the properties of matrix arithmetic and the transpose.

Inverse Matrices and Elementary Matrices  Here we’ll define the inverse and take a look at some of its properties.  We’ll also introduce the idea of Elementary Matrices.

Finding Inverse Matrices  In this section we’ll develop a method for finding inverse matrices.

Special Matrices  We will introduce Diagonal, Triangular and Symmetric matrices in this section.

LU-Decompositions  In this section we’ll introduce the LU-Decomposition as a way of “factoring” certain kinds of matrices.

Systems Revisited  Here we will revisit solving systems of equations.  We will take a look at how inverse matrices and LU-Decompositions can help with the solution process.  We’ll also take a look at a couple of other ideas in the solution of systems of equations.