TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Applying an elementary row operation is left multiplication by its elementary matrix

Statement

Let $A\in M_{n\times p}(F)$ and let $E\in M_n(F)$ be the elementary matrix of an elementary row operation. Then $EA$ is exactly the matrix obtained by applying that row operation to $A$ .

Facts & Assumptions

Given: A field $F$ , a matrix $A\in M_{n\times p}(F)$ and an elementary matrix $E\in M_n(F)$ .

[L1]

An elementary matrix is obtained by applying one elementary row operation to $I_n$ (Elementary matrices obtained by applying one elementary row operation to an identity matrix).

[L2]

Matrix multiplication is given by $(EA)_{ij}=\sum_{k<n}e_{ik}a_{kj}$ , and $I_n$ has entries $1$ on the diagonal and $0$ elsewhere (Rectangular matrix multiplication and the identity matrix $I_n$ , including zero-sized shapes).

Proof

technique · direct

1.1

If $E$ swaps rows $r,s$ of $I_n$ , the displayed sum exchanges rows $r,s$ of $A$ ; if $E$ scales row $r$ of $I_n$ by $c$ , it replaces row $r$ of $A$ by $cR_r$ ; and if $E$ adds $c$ times row $s$ to row $r$ of $I_n$ , it replaces row $r$ of $A$ by $R_r+cR_s$ , leaving every other row fixed.

L1L2algebra

2.1

These are precisely the three corresponding row operations on $A$ , so left multiplication by $E$ performs the operation. The assertion is vacuous when there is no row on which an elementary operation can be defined, and it remains valid when $p=0$ .

step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 7 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. Three, §IV.3 (standard reference, not scraped)