LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every elementary row operation has an elementary inverse, so row equivalence is an equivalence relation

Statement

Every elementary row operation on a finite matrix over a field has an elementary inverse. Consequently row equivalence is reflexive, symmetric and transitive.

Facts & Assumptions

Given: A field $F$ and finite matrices of one fixed shape over $F$ .

[L1]

An elementary row operation is a row swap, a scaling of one row by a nonzero field element, or addition of a scalar multiple of one row to another (Elementary row operations and row equivalence for finite matrices over a field).

Proof

technique · direct

1.1

A swap is its own inverse; the inverse of $R_{p} \leftarrow c R_{p}$ is $R_{p} \leftarrow c^{- 1} R_{p}$ because $c \neq 0$ in a field; and the inverse of $R_{p} \leftarrow R_{p} + c R_{q}$ is $R_{p} \leftarrow R_{p} - c R_{q}$ .

L1algebra

2.1

The empty operation sequence proves reflexivity, reversing a sequence and replacing every operation by its inverse proves symmetry, and concatenating two finite sequences proves transitivity. These arguments also cover empty matrices and the empty reduction.

step 1.1∎

Depends on

Elementary row operations and row equivalence for finite matrices over a field

Used by

Dependency tree · two levels

3 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

J. Hefferon, Linear Algebra, 4th ed., Ch. One, §III.1 (standard reference, not scraped)