MathematicsChapter 2: Matrices and Determinants

Linear Systems: Consistent Or Not

A system Ax = b. Consistent: at least one solution.

Learning objectives

State the main ideas and notation used in linear systems: consistent or not.
Apply the rules of linear systems: consistent or not in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.

Mathematical form

Ax = b. Consistent

Ax=0 always has the trivial x

det=0. Meaning

z = 4

Where / symbols

LINEAR SYSTEMS: CONSISTENT OR NOT: A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.
1. THREE CASES: Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.
2. HOMOGENEOUS: 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).

Relationship

consistent = at least one x
0 = 1 means inconsistent

Overview

What the last rows say:

Three Cases

Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

Examples Unique · Free variable · Contradiction

Homogeneous

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

WHAT — LINEAR SYSTEMS: CONSISTENT OR NOT

RULE — 1. THREE CASES

Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

METHOD — 2. HOMOGENEOUS

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).

WORKED EXAMPLE

Worked tell: after elimination you see 0 0 1 | 4. that is z = 4, fine if z is an unknown but 0 0 0 | 4 means 0=4, inconsistent read the bar: coefficients | right-hand side

How to answer a linear systems: consistent or not question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

Worked tell: after elimination you see 0 0 1 | 4.
that is z = 4, fine if z is an unknown
but 0 0 0 | 4 means 0=4, inconsistent
read the bar: coefficients | right-hand side

In simple words

Homogeneous is never inconsistent. Non-homogeneous can be.

Key points

consistent = at least one x
0 = 1 means inconsistent
pivots vs unknowns
always consistent

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.