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

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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 $E A$ 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 $(E A)_{i j} = \sum_{k < n} e_{i k} a_{k j}$ , 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 $c R_{r}$ ; and if $E$ adds $c$ times row $s$ to row $r$ of $I_{n}$ , it replaces row $r$ of $A$ by $R_{r} + c R_{s}$ , 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 · two levels

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Sources

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