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Lesson 17

Gauss-Jordan Elimination to RREF

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In our last session, we mastered Gaussian elimination, a systematic process for transforming an augmented matrix into row echelon form (REF). This simplified the matrix into a "staircase" structure, allowing us to find the solution using back substitution. While effective, that final step of back substitution can still involve a fair bit of calculation.

This lesson introduces a more powerful extension of this method: Gauss-Jordan elimination. By adding a second phase of row operations—the "backward steps"—we can refine the matrix even further into reduced row echelon form (RREF). The beauty of this form is that it makes the solution to the system immediately visible, eliminating the need for back substitution entirely. By the end of this lesson, you will be able to take any augmented matrix and transform it all the way to its unique reduced row echelon form.

Beyond the Staircase: Reduced Row Echelon Form

In the previous lesson, we stopped at row echelon form (REF). This form is defined by three rules:

  1. All-zero rows are at the bottom.
  2. The leading entry (pivot) of any row is to the right of the pivots in the rows above it.
  3. All entries in a column below a pivot are zero.

Reduced row echelon form (RREF) must satisfy these three conditions, plus two more, making it an even more organized and "solved" state.

[PDF] (Reduced) Row Echelon Form

This document from Georgia Tech's math department provides a concise definition of RREF, building directly upon the definition of REF.

Read the beginning of the document to understand the five conditions for a matrix to be in RREF. Pay close attention to the two additional requirements that distinguish it from REF, found in points 4 and 5 of the full definition.

To summarize, a matrix is in reduced row echelon form (RREF) if it meets all the conditions for REF, and:
4. The pivot in each nonzero row is 1 (we called this a "leading 1" in the last lesson).
5. Each leading 1 is the only nonzero entry in its column.

This fifth condition is the key addition. It means we must not only have zeros below each pivot, but also above them.

Here is a comparison:

Row Echelon Form (REF):

Notice the entries above the pivots in the second and third columns (the 2, 3, and 5) can be non-zero.

Reduced Row Echelon Form (RREF):

Here, the pivot in each row is the only non-zero number in its entire column. This clean, diagonal structure is our goal.

The Algorithm: From Gaussian to Gauss-Jordan

Gauss-Jordan elimination is a two-phase process:

  1. Forward Phase (Gaussian Elimination): Use elementary row operations to put the matrix into row echelon form (REF). This is exactly what we did in the previous lesson.
  2. Backward Phase: Starting from the REF, use more elementary row operations to create zeros above each pivot.

The backward phase works as follows:

  • Start with the last pivot (the one in the bottom-most non-zero row).
  • Use row replacement operations to make all entries above this pivot zero.
  • Move to the next pivot up and repeat the process.
  • Continue until all entries above all pivots are zero.

The following video demonstrates this "backward" part of the process very clearly. It picks up right where the last lesson left off: with a matrix in row echelon form.

Gaussian Elimination and Gauss Jordan Elimination (Gauss Elimination Method)

In "Gaussian Elimination and Gauss Jordan Elimination," BriTheMathGuy first performs Gaussian elimination and then explicitly shows the extra steps needed for Gauss-Jordan.

Watch the segment from "if you want to do Gauss-Jordan". The presenter starts with a matrix already in REF and methodically applies row operations to create zeros above the pivots, transforming it into RREF and showing how to read the final answer.

A Complete Worked Example

Let's put both phases together and see the entire journey from a system of equations to its solution via RREF. The following image provides a fantastic overview of this complete process.

This diagram illustrates the full Gauss-Jordan elimination pipeline. A system of equations is converted into an augmented matrix. Elementary row operations are first used to achieve Row Echelon Form (the "forward phase"), and then more operations are used to clear the entries above the pivots, resulting in Reduced Row Echelon Form (the "backward phase").

Now, let's watch a detailed, step-by-step application of the full algorithm. This video will walk you through every row operation needed to solve a 3x3 system.

Gauss Jordan Elimination & Reduced Row Echelon Form

This video, "Gauss Jordan Elimination & Reduced Row Echelon Form," provides a very clear and methodical walkthrough of a complete problem.

Watch the entire video from start to finish. As you watch, try to identify the two main phases: Forward Phase: Notice how the presenter first focuses on creating the staircase of pivots with zeros below them (from roughly the first few operations). This gets the matrix into a state resembling REF. Backward Phase: Observe how the strategy then shifts to creating zeros above the pivots, working from right to left (from about clearing the upper entries). Final Answer: See how the final RREF matrix at the end allows the values of x, y, and z to be read directly.

The reason Gauss-Jordan elimination is so powerful is precisely because of that final step. Once the augmented matrix is in the form [ I | **d** ], where I is the identity matrix, the system of equations becomes:

The solution is simply , , and . The solution vector d is sitting right there in the augmented column.

Conclusion

In this lesson, we completed our journey of algorithmic solutions for linear systems by introducing Gauss-Jordan elimination. By adding a "backward phase" to the Gaussian elimination we learned previously, we can arrive at the maximally simplified reduced row echelon form.

Here are the main takeaways:

  • Reduced Row Echelon Form (RREF) is a strict form of a matrix where each pivot (leading 1) is the only non-zero entry in its column.
  • Gauss-Jordan Elimination is a two-phase algorithm that brings a matrix to RREF: the forward phase creates zeros below the pivots (achieving REF), and the backward phase creates zeros above them.
  • The primary advantage of RREF is that for a system with a unique solution, the solution can be read directly from the augmented column, eliminating the need for back substitution.

A fascinating property of RREF is its uniqueness: any matrix is row equivalent to one and only one matrix in reduced row echelon form. No matter what sequence of valid row operations you perform, you will always arrive at the same RREF.

So far, we have focused on systems that have a nice, unique solution. But what happens if the system has no solution, or infinitely many? In our next lesson, we will learn how to interpret the row echelon form of a matrix to determine exactly which of these three cases we are dealing with.