Alphabeta Math

Gaussian Elimination and Row Reduction: Examples and Counterexamples

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✓ 9 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

A 3×3 system row reduces to a unique solution

Example

A 3×3 system has no solution because its augmented column contains a pivot

Example

A consistent underdetermined system has an affine two-parameter solution set

Example

A 3×3 inverse computed by row reducing [A∣ I]

Example

One matrix has two distinct row echelon forms but one reduced row echelon form

Example

The rank and solution behaviour of a parameterised matrix change at one exceptional parameter

Example

An invertible 3×3 real matrix factored explicitly into elementary matrices

Counterexample

Row-equivalent matrices can have different column spaces even though their column ranks agree

Counterexample

Over F₂, the equation x+y=0 has exactly two solutions, so the infinite-field hypothesis is necessary

Sources