ExampleConstruction: AI-generatedVerification: 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.

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

$A = (\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 1 & 3 \end{array}), B = (\begin{array}{cc} 1 & 0 \\ 2 & 1 \\ 0 & 1 \end{array}) .$

The product $A B$ is $2$ by $2$ , the product $B A$ 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, $tr (A B) = tr (B A)$ (For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ ).

Verification

technique · direct

1.1

Direct multiplication gives $A B = (\begin{array}{cc} 5 & 2 \\ 2 & 4 \end{array})$ and $B A = (\begin{array}{ccc} 1 & 2 & 0 \\ 2 & 5 & 3 \\ 0 & 1 & 3 \end{array})$ .

givenL1

2.1

Hence $tr (A B) = 5 + 4 = 9$ and $tr (B A) = 1 + 5 + 3 = 9$ 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 · two levels

5 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.

A 22 by 33 matrix and a 33 by 22 matrix give square products of different sizes but equal traces