On this page, we discuss matrices in reduced row-echelon form (RREF), and how to use row operations to bring the augmented matrix of a SLE into this form, using the methods of Gaussian Elimination or Gauss-Jordan Elimination.
Note: The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to "read ahead" before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Recognize whether a given matrix is in row echelon form, reduced row echelon form, or neither.
- Construct solutions to linear systems whose corresponding augmented matrices are in row echelon form or reduced row echelon form.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use Gaussian elimination to find the general solution of a linear system.
- Use Gauss-Jordan elimination to find the general solution of a linear system.
- Analyze a homogeneous linear systems.
To prepare for class
Reduced Row-Echelon Form (RREF)
Watch this short video which shows the difference between a row-echelon form and a reduced row-echelon form (RREF):
Watch this video which explains (with several detailed examples) how to recognize the number of solutions from the RREF, and how to write down the solutions in parametric form when there are infinitely many solutions:
Gaussian Elimination and Gauss-Jordan Elimination
Watch this video which shows the method of Gaussian Elimination with "Back-Substitution" on the example of a \(3\times 3\) SLE (with unique solution):
Watch this video which shows the method of Gauss-Jordan Elimination on the example of a \(3\times 3\) SLE (with unique solution):
Watch this video which shows the geometric effect on the planes when using Gauss-Jordan Elimination on a \(3\times 3\) SLE (with unique solution):
Gauss-Jordan with Infinitely many solutions
Watch this video which shows the method of Gauss-Jordan Elimination on the example of a \(2\times 3\) SLE (with infinitely many solutions):
Watch this video which shows the method of Gaussian Elimination on the example of a \(3\times 3\) SLE (with infinitely many solutions) - and which also shows a neat (and highly recommended) trick of using a row-checksum to double-check your calculations:
Watch this video which shows the method of Gaussian Elimination on the example of a \(4\times 6\) SLE (with infinitely many solutions) - and which also shows how to write the solutions in parametric vector form:
Online RREF Calculators
There are many websites which offer a "RREF Calculator", some even showing step-by-step solutions. Most of these have the disadvantage that they only use the "official", straight-forward application of the Gauss-Jordan algorithm - which is not always the fastest or nicest (which often creates nasty fractions which could otherwise be avoided until much later).
The following website seems to be doing a much better job at showing a "smart" sequence of row operations in most cases:
Note: in the step-by-step solutions on this website, they show the operation of "adding a multiple of a row to another" in 3 separate steps: by first multiplying the row by the number, then adding to the other row - and then "restoring" the original row. You should not write it out on paper like that - this is only done here so you can better see the steps and the calculations.
Finally, remember that you should use this (and other websites) wisely, for example to double-check your work, or if you're really stuck - don't just use it to get "free" marks on your WebWork, otherwise you'll be very lost in the exam... Or as somebody else would say it: