Topic: Systems and Matrices

Subtopic: Gaussian Elimination

**Overview**

Large linear systems take far too much time to solve by substitution or elimination, but matrix methods are efficient and easy to adapt to electronic solvers. In today's lesson we express linear systems in augmented matrices and solve them using matrix methods. We use matrix row operations to transform the system to REF form (a diagonal of ones with a triangle of zero below the diagonal) and use back-substitution to solve the system. This is the __Gaussian Elimination method__. Or we could go further using matrix row ops to get a second triangle of zeros above the diagonal of ones (RREF) and reading the answers straight from the resulting matrix. This is __Gauss-Jordan method__. REF and RREF can be conducted electronically, but first be sure you can algebraically transform a matrix to REF and RREF forms.

To determine if a system is consistent, inconsistent, or dependent use RREF electronically and note that:

- If the RREF form has a row containing zeros on the left of the vertical bar and a nonzero number on the right of the vertical bar, then that indicates an inconsistent system (i.e., no solution).
- If the RREF form has a row containing all zeros, then that indicates a dependent system (i.e., an infinite number of solutions) and the solution should be written in terms of an appropriate parameter.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 5.3.1 Understand and be able to properly use basic matrix terminology
- 5.3.2 Translate a linear system to an augmented matrix
- 5.3.3 Recognize a matrix as being in REF or in RREF form
- 5.3.4 Use matrix row operations to transform the augmented matrix of a 3x3 or 4x4 linear system to REF form
- 5.3.5 From REF form perform back-substitution to solve the original 3x3 or 4x4 linear system
- 5.3.6 Algebraically solve a 3x3 or 4x4 linear system by the Gaussian Elimination method
- 5.3.7 Use matrix row operations to transform the REF form matrix of a 3x3 or 4x4 linear system to RREF form
- 5.3.8 Algebraically solve a 3x3 or 4x4 linear system by the Gauss-Jordan method
- 5.3.9 Electronically transform a 3x3 or higher linear system to RREF form including entering the matrix, transforming to RREF form, and recognizing the answers
- 5.3.10 Recognize if a system is consistent, inconsistent, or dependent by an electronic RREF solver
- 5.3.11 Correctly write the solution to a dependent system
- 5.3.12 Solve a variety of applications of linear systems including consistent, inconsistent, and dependent. Concentrate on setting up the system of equations then solve them electronically.

**Terminology**

Define: matrix, augmented matrix, row echelon form (REF), reduced row echelon form (RREF), back-substitution, consistent system (one solution), inconsistent system (no solution), dependent system (an infinite number of solutions)

**Text Notes**

When performing Gaussian Elimination method some texts force the diagonals of ones before getting the lower triangle of zeros. Personally I recommend that you get the lower triangle of zeros before forcing the diagonal of ones. This will delay having to work with fractions as long as possible.

**Supplementary Resources**

If you need assistance with the RREF feature of your calculator see the supplementary resources in Algebra I Lesson Notes #5.6.

If you don't have a handheld graphing calculator then RREF can be performed using the **Online Row Reducer**. ⇐ Useful!