Title: How to Use Gauss-Jordan Elimination Step-by-Step – RREF Made Easy

This lesson shows how to perform Gauss-Jordan Elimination to reduce a system of equations into Reduced Row-Echelon Form (RREF). This is a continuation from the prior Gaussian Elimination example but takes it further to fully solve for each variable.

Matrix Setup (from previous work)

Starting augmented matrix:

[ 1  -2   3 | -2 ]
[ 0   3  -5 |  6 ]
[ 0   0   0 |  1 ]

Since the third row is [ 0 0 0 | 1 ], we already concluded this is inconsistent (no solution).

But we continue Gauss-Jordan to learn the process.

Gauss-Jordan Elimination Strategy:

Use counterclockwise elimination:

1. Eliminate below pivot a11

2. Then below a22

3. Then below a33

4. Then eliminate above each pivot

5. Finally, divide rows to get leading 1s

Step-by-Step:

1. Multiply row 2 by 3/5
   → 3/5 * R2 + R1 → R1

2. Result:

[ 1   9/5   2   8/5 ]
[ 0     3  -5     6 ]
[ 0     0   0     1 ]

1. Now eliminate back into R1 using R3:
   → -6*R3 + R2, etc.

Eventually, you reach:

[ 1  0  0 | -1/3 ]
[ 0  1  0 |  5/3 ]
[ 0  0  1 |    1 ]

Final Result:

This is Reduced Row-Echelon Form (RREF). But since the matrix had [ 0 0 0 | 1 ] earlier, it’s still inconsistent.

Key Takeaway:

Gauss-Jordan Elimination gives you full control over the system and reveals inconsistencies. If you’re asked to fully reduce a matrix into RREF, this is the path to follow.

This tutorial is part of Chapter 9: Gauss-Jordan Elimination from the Ultimate Crash Course for STEM Majors. It’s designed to help you understand how matrix algebra is used to prove solutions exist—or not.

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