TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system

Statement

If an elementary row operation transforms the augmented matrix $[A ∣ b]$ into $[A^{'} ∣ b^{'}]$ , then $S (A, b) = S (A^{'}, b^{'}) .$ Consequently every finite row reduction of an augmented matrix preserves its solution set exactly.

Facts & Assumptions

Given: A system $A x = b$ over a field and one elementary row operation applied to its augmented matrix.

[L1]

Elementary row operations are swaps, nonzero row scalings and additions of a multiple of one row to another (Elementary row operations and row equivalence for finite matrices over a field).

[L2]

Every elementary row operation has an elementary inverse (Every elementary row operation has an elementary inverse, so row equivalence is an equivalence relation).

[L3]

The system $A x = b$ consists of the row equations $\sum_{j < n} a_{i j} x_{j} = b_{i}$ , and $[A ∣ b]$ records their coefficients and right-hand sides (Matrix equation $A x = b$ , its solution set, consistency, homogeneous systems and the augmented matrix $[A ∣ b]$ ).

Proof

technique · direct

1.1L1L3algebra

A row swap only reorders equations; multiplying an equation by $c \neq 0$ preserves it in both directions after multiplication by $c^{- 1}$ ; and replacing one equation by itself plus $c$ times another preserves simultaneous satisfaction because the added equation already holds. Thus every solution of the original system solves the transformed system.

2.1step 1.1L2∎

Apply the same argument to the inverse operation from [L2] to obtain the reverse inclusion. Hence the solution sets are equal, without any assumption that $A$ is square or invertible; iteration gives the finite-reduction claim.

Depends on

Dependency tree · 2 levels

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §1.2