ExampleConstruction: AI-generatedVerification: AI-generatedSession-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.

A $2$ by $3$ matrix and a $3$ by $2$ matrix give square products of different sizes but equal traces

Example

Over any field $F$ , let

$A=\begin{pmatrix}1&2&0\\0&1&3\end{pmatrix},\qquad B=\begin{pmatrix}1&0\\2&1\\0&1\end{pmatrix}.$

The product $AB$ is $2$ by $2$ , the product $BA$ is $3$ by $3$ , and their traces are equal.

Facts & Assumptions

Given: The displayed rectangular matrices over a field $F$ .

[L1]

For conformable rectangular matrices, $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ (For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$ , $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ ).

Verification

technique · direct

1.1

Direct multiplication gives $AB=\begin{pmatrix}5&2\\2&4\end{pmatrix}$ and $BA=\begin{pmatrix}1&2&0\\2&5&3\\0&1&3\end{pmatrix}$ .

givenL1

2.1

Hence $\operatorname{tr}(AB)=5+4=9$ and $\operatorname{tr}(BA)=1+5+3=9$ as elements of $F$ , verifying [L1] for products of different square sizes.

step 1.1L1∎

Depends on

For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\operatorname{tr}(AB)=\operatorname{tr}(BA)$

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 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.

A 222 by 333 matrix and a 333 by 222 matrix give square products of different sizes but equal traces