DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Elementary matrices obtained by applying one elementary row operation to an identity matrix

Definition

Let $F$ be a field and $n \in N$ . An elementary matrix is a matrix $E \in M_{n} (F)$ obtained by applying one elementary row operation (Elementary row operations and row equivalence for finite matrices over a field) to the identity matrix $I_{n}$ (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

Accordingly there are three types: $E_{p q}$ interchanges rows $p$ and $q$ ; $D_{p} (c)$ multiplies row $p$ by $c \neq 0$ ; and $T_{p q} (c)$ adds $c$ times row $q$ to the distinct row $p$ . When $n = 0$ there is no elementary matrix because there is no row on which to perform an operation.

Depends on

Used by

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Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.3 (standard reference, not scraped)