ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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A $3 \times 3$ system has no solution because its augmented column contains a pivot

Example

The real system $x + y + z = 1, 2 x + 2 y + 2 z = 3, x - y = 0$ is inconsistent.

Facts & Assumptions

Given: The displayed system and its augmented matrix.

[L1]

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

[L2]

Consistency is equivalent to equality of coefficient and augmented ranks (The system $A x = b$ is consistent if and only if $rank (A) = rank ([A ∣ b])$ ).

Verification

technique · direct

1.1

Subtracting twice row $0$ from row $1$ gives the row $(0, 0, 0 ∣ 1)$ , which normalizes to a pivot in the augmented column and represents $0 = 1$ .

L1algebra

2.1

The coefficient matrix has rank $2$ because rows $(1, 1, 1)$ and $(1, - 1, 0)$ are independent, while the augmented matrix has the additional pivot and rank $3$ . Thus [L2] independently gives inconsistency.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §1.3 (standard reference, not scraped)

A 3×33×3 system has no solution because its augmented column contains a pivot