ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

An invertible $3 \times 3$ real matrix factored explicitly into elementary matrices

Example

The matrix $A = (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array})$ has the elementary factorisation $A = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}) (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}) .$

Facts & Assumptions

Given: The displayed real matrix $A$ .

[L1]

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

[L2]

A reduction $E_{r} \dots E_{1} A = I$ reverses to an elementary factorisation (A finite row reduction from $A$ to $B$ is encoded by $B = E_{r} \dots E_{1} A$ ).

Verification

technique · direct

1.1

Subtract row $2$ from row $1$ , then subtract the new row $1$ from row $0$ ; these elementary operations reduce $A$ to $I_{3}$ , so their matrices satisfy $E_{2} E_{1} A = I_{3}$ .

L1L2algebra

2.1

Reversing the operations gives the two displayed row-addition matrices. Their direct product is $A$ , verifying the factorisation entry by entry.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.3 (standard reference, not scraped)

An invertible 3×33×3 real matrix factored explicitly into elementary matrices