Lesson 18.3

Gauss-Jordan Elimination

Continuing to reduced row-echelon form where solutions appear directly.

Introduction

Gaussian elimination gets us to row-echelon form, but we still need back-substitution to find the answers.Gauss-Jordan elimination continues the process—creating zeros above each pivot and making each pivot equal to 1. The result is reduced row-echelon form (RREF), where solutions appear directly in the last column.

1

Prerequisite Connection

You can reduce a matrix to row-echelon form using Gaussian elimination.

2

Today's Increment

We continue to reduced row-echelon form (RREF) where pivots = 1 and zeros are above AND below.

3

Why This Matters

RREF is the canonical form—every matrix has exactly one RREF. Calculators and computers report this form for consistency.

Key Concepts

Reduced Row-Echelon Form (RREF)

A matrix is in RREF when:

  • It's in row-echelon form (REF)
  • Every pivot is exactly 1
  • Every entry ABOVE each pivot is also 0

Reading Solutions from RREF

The identity matrix on the left means the right column IS the solution:

Free Variables

Columns without pivots give free variables. Set them to parameters for infinitely many solutions.

Worked Examples

Example 1: REF to RREF (Basic)

Convert to RREF:

Step 1: Make pivot in row 2 equal to 1

Step 2: Eliminate above pivot

Solution:

Example 2: Three Variables (Intermediate)

Reduce to RREF:

Step 1: Scale pivots to 1

and

Step 2: Eliminate above pivots

, then , then

Solution:

Example 3: Infinitely Many Solutions (Advanced)

Find all solutions:

Step 1: Eliminate above pivot

Step 2: Identify free variable

Column 2 has no pivot → is free. Let

Solution:

where

Common Pitfalls

Stopping at REF instead of RREF

RREF requires zeros ABOVE pivots too. If you still need back-substitution, you're not done.

Missing free variables

If a column has no pivot, that variable is FREE—introduce a parameter and express other variables in terms of it.

Forgetting to scale pivots to 1

RREF requires each pivot to be exactly 1, not just nonzero.

Real-World Application

3D Graphics Transformations

Video games and CGI use RREF to find transformation matrices that map one set of points to another. Need to rotate, scale, and translate a 3D model? Set up the equations for where vertices should go, reduce to RREF, and extract the transformation parameters directly.

RREF gives unique, canonical solutions—essential for consistent rendering across different graphics engines.

Practice Quiz

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