Lesson 18.2

Gaussian Elimination and Row-Echelon Form

Using row operations to transform matrices into a staircase pattern for solving systems.

Introduction

Once we have an augmented matrix, we need a systematic way to solve it. Gaussian elimination uses three simple row operations to create zeros below each pivot (leading nonzero entry). The result isrow-echelon form—a staircase pattern where back-substitution reveals all the solutions.

1

Prerequisite Connection

You can write a system as an augmented matrix and understand matrix dimensions.

2

Today's Increment

We apply the three elementary row operations to reach row-echelon form (REF).

3

Why This Matters

Gaussian elimination is the algorithm inside every computer solving linear systems—from spreadsheets to physics simulations.

Key Concepts

The Three Elementary Row Operations

1. Swap: Exchange two rows

2. Scale: Multiply a row by a nonzero constant

3. Replace: Add a multiple of one row to another

Row-Echelon Form (REF)

A matrix is in REF when:

  • All zero rows are at the bottom
  • Each pivot is to the RIGHT of the pivot in the row above
  • All entries BELOW each pivot are zero

Back-Substitution

Once in REF, solve from the bottom row up. The last equation gives one variable directly; substitute upward to find the rest.

Worked Examples

Example 1: Two Equations (Basic)

Solve using Gaussian elimination:

Step 1: (get 1 in top-left)

Step 2:

Back-substitute:

Solution:

Example 2: Three Variables (Intermediate)

Reduce to REF:

Step 1: and

Step 2:

Now in Row-Echelon Form!

Example 3: No Solution (Advanced)

Analyze this system:

Apply

Row 2 says — contradiction!

The system has NO SOLUTION (inconsistent).

Common Pitfalls

Forgetting to apply operations to the ENTIRE row

When doing , you must subtract from EVERY entry in row 2, including the constant.

Multiplying a row by zero

Row operations must use NONZERO constants for scaling. Multiplying by zero destroys information.

Arithmetic errors in multi-step operations

Double-check each computation. One sign error early on cascades through the entire solution.

Real-World Application

Circuit Analysis (Kirchhoff's Laws)

Electrical engineers use Gaussian elimination to solve circuits. Each loop in a circuit gives one equation (sum of voltage drops = 0), and each node gives another (current in = current out). For complex circuits with dozens of components, this becomes a large system of linear equations—perfect for matrix methods.

A circuit with 10 unknown currents requires solving 10 simultaneous equations—Gaussian elimination handles this systematically.

Practice Quiz

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