MathematicsChapter 2: Matrices and Determinants
Homogeneous Linear Systems
Ax=0. Always x=0. Non-trivial kernel if rank < n.
Learning objectives
- State the rule or meaning of homogeneous linear systems in easy words.
- Apply homogeneous linear systems in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
Formula
Ax=0. Always x
Where / symbols
- HOMOGENEOUS LINEAR SYSTEMS
- Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
- HOMOGENEOUS
- Ax=0.
Relationship
- consistent = at least one x
- 0 = 1 means inconsistent
Overview
Understand the idea, then apply it:
Main Idea
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
Examples Trivial solution · Free variables if singular · Null space named
Exam Use
Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — HOMOGENEOUS LINEAR SYSTEMS
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
WHY — Why it matters
This rule is a separate learning unit inside linear systems: consistent or not. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a homogeneous linear systems question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
Homogeneous is never inconsistent. Non-homogeneous can be.
Key points
- always consistent
- consistent = at least one x
- 0 = 1 means inconsistent
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Homogeneous Linear Systems
Sticky Notes · Quick Revision
Core Idea
- Ax=0.
What to Know
- Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
Example or Evidence
- Trivial solution
- Free variables if singular
- Null space named
How to Answer
- Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- Homogeneous is never inconsistent. Non-homogeneous can be.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- Ax=0.
- Trivial solution
- Free variables if singular
- State the rule, show the working, and check the answer.
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