Unit 3
Elementary Matrices and the Matrix Inverse

In Unit 1, you studied elementary row operations as a method for solving systems of linear equations. Recall that the three elementary row operations are multiplying a row by a nonzero constant, interchanging two rows, and adding a nonzero multiple of one row to another row.

In Unit 2, you learned some basic elements of matrix arithmetic, and the definition of the inverse of a matrix. You also looked at some simple theorems and their proofs.

In this unit, you will combine these skills to prove a basic theorem relating properties of matrices and solutions of systems of linear equations. Some of the reasoning in this unit is very abstract. This does not mean that it is difficult, only that you will have to be careful, and go one step at a time. Following a mathematical argument is like putting up a brick wall. Each step in the argument is a brick that must be fitted carefully into the structure at the right time and in the right place. Often, it helps to try to visualize what is being presented in terms of simple examples and diagrams. Working out simple numerical examples will improve your computational and logical skills. Making rough sketches can help you to get an overview, much like looking at the floor plan of a building. These are skills that improve with practice, so you should not feel discouraged if this kind of work seems difficult at first. And remember, you can count on your tutor to help you through the really hard parts!

Objectives

When you have completed this unit, you should be able to

  1. define the term “elementary matrix,” and state the relationship between elementary matrices and elementary row operations.
  2. explain what it means for one matrix to “row equivalent” to another matrix.
  3. state a theorem relating invertibility, row equivalence to the identity matrix, and solutions of systems of linear equations.
  4. compute the inverse of any invertible matrix A, and use this inverse to solve any system of equations for which A is the coefficient matrix.
  5. use Gaussian elimination to determine what conditions must be satisfied in order for an arbitrary system of linear equations to have a solution.