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

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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

1.1

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.

L1L3algebra

2.1

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.

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

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Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §1.2 (standard reference, not scraped)