DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Elementary matrices obtained by applying one elementary row operation to an identity matrix

Definition

Let $F$ be a field and $n\in\mathbb N$ . 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: $E_{pq}$ interchanges rows $p$ and $q$ ; $D_p(c)$ multiplies row $p$ by $c\ne0$ ; and $T_{pq}(c)$ 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

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 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)