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

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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.
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Every elementary matrix is invertible, with inverse given by the reverse elementary operation

Statement

Every elementary matrix $E \in M_{n} (F)$ is invertible. Its inverse is the elementary matrix belonging to the inverse row operation.

Facts & Assumptions

Given: An elementary matrix $E$ corresponding to a row operation $ρ$ .

[L1]

The operation $ρ$ has an elementary inverse $ρ^{- 1}$ (Every elementary row operation has an elementary inverse, so row equivalence is an equivalence relation).

[L2]

Applying an elementary row operation is left multiplication by its elementary matrix (Applying an elementary row operation is left multiplication by its elementary matrix).

[L3]

A square matrix is invertible when it has a two-sided inverse (Invertible matrices and the general linear group ${GL}_{n} (F)$ ).

Proof

technique · direct

1.1L1L2

Let $E^{'}$ be the elementary matrix of $ρ^{- 1}$ . Applying $ρ$ and then $ρ^{- 1}$ to $I_{n}$ gives $E^{'} E = I_{n}$ , while applying them in the reverse order gives $E E^{'} = I_{n}$ .

2.1step 1.1L3∎

Thus $E^{'}$ is a two-sided inverse of $E$ , so $E$ is invertible and $E^{- 1} = E^{'}$ .

Depends on

Dependency tree · 2 levels

Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.3