Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis
S = contains supplemental resources
Course: College Algebra
Topic: Systems and Matrices
Subtopic: Gaussian Elimination


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:


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


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!