Alphabeta MathHow 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.
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- 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 elementary row operation has an elementary inverse, so row equivalence is an equivalence relation
Statement
Every elementary row operation on a finite matrix over a field has an elementary inverse. Consequently row equivalence is reflexive, symmetric and transitive.
Facts & Assumptions
Given: A field and finite matrices of one fixed shape over .
An elementary row operation is a row swap, a scaling of one row by a nonzero field element, or addition of a scalar multiple of one row to another (Elementary row operations and row equivalence for finite matrices over a field).
Proof
A swap is its own inverse; the inverse of is because in a field; and the inverse of is .
The empty operation sequence proves reflexivity, reversing a sequence and replacing every operation by its inverse proves symmetry, and concatenating two finite sequences proves transitivity. These arguments also cover empty matrices and the empty reduction.