CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The system $Ax=b$ is consistent if and only if $\operatorname{rank}(A)=\operatorname{rank}([A\mid b])$

Statement

For every finite system $Ax=b$ over a field, $Ax=b\text{ is consistent}\quad\Longleftrightarrow\quad \operatorname{rank}(A)=\operatorname{rank}([A\mid b]).$

Facts & Assumptions

Given: A coefficient matrix $A$ and augmented matrix $[A\mid b]$ .

[L1]

Consistency is equivalent to absence of a pivot in the augmented column of the RREF (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

Matrix rank equals the number of pivots (Row rank equals column rank, and both equal the number of pivots).

Proof

technique · direct

1.1

Row reduce the augmented matrix. Its coefficient block is an echelon form row equivalent to $A$ , while the whole reduced matrix is an echelon form of $[A\mid b]$ . Thus [L2] identifies their respective pivot counts with the two ranks. Those counts differ exactly when the augmented column supplies one additional pivot.

L1L2

2.1

By [L1], absence of that additional augmented pivot is equivalent to consistency. Step 1.1 therefore proves both directions.

step 1.1L1∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. One, §III.2 (standard reference, not scraped)

The system Ax=bAx=bAx=b is consistent if and only if rank⁡(A)=rank⁡([A∣b])\operatorname{rank}(A)=\operatorname{rank}([A\mid b])rank(A)=rank([A∣b])