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.

Every invertible finite square real matrix is a finite product of elementary matrices

Statement

Every invertible matrix $A\in M_n(\mathbb R)$ is a finite product of elementary matrices. The identity matrix is represented by the empty product.

Facts & Assumptions

Given: An invertible real matrix $A\in M_n(\mathbb R)$ .

[L1]

An invertible matrix has RREF $I_n$ (Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent).

[L2]

A row reduction is left multiplication by a product of its elementary matrices (A finite row reduction from $A$ to $B$ is encoded by $B=E_r\cdots E_1A$ ).

[L3]

The inverse of an elementary matrix is elementary (Every elementary matrix is invertible, with inverse given by the reverse elementary operation).

[L4]

The real numbers form a field (The reals form a field).

Proof

technique · constructive

1.1

By [L1] row reduction takes $A$ to $I_n$ ; by [L2] there are elementary real matrices with $E_r\cdots E_1A=I_n$ .

L1L2L4construct

2.1

Multiply by the explicit inverses in reverse order to obtain $A=E_1^{-1}\cdots E_r^{-1}$ , and each factor is elementary by [L3].

step 1.1L3algebra

3.1

If $A=I_n$ , take $r=0$ and the empty product. No determinant is used.

step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 10 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)