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.

An integer matrix of determinant $2$ is invertible over $\mathbb Q$ but not over $\mathbb Z$

Example

The integer matrix $A=\operatorname{diag}(2,1)$ has determinant $2$ . It is invertible as a matrix over $\mathbb Q$ , but it is not invertible as a matrix over $\mathbb Z$ .

Facts & Assumptions

Given: $A=\begin{pmatrix}2&0\\0&1\end{pmatrix}$ .

[L1]

The $2\times2$ determinant is $ad-bc$ (The Leibniz formula gives $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$ ).

[L2]

An invertible matrix over a commutative ring has unit determinant (An invertible square matrix over a commutative ring has unit determinant).

[L3]

The rationals form a field (The rationals form a field).

[L4]

The integers form a commutative ring (The integers form a commutative ring).

[L5]

The only units of $\mathbb Z$ are $1$ and $-1$ ( $(\mathbb{Z}, \cdot, 1)$ is a commutative monoid whose group of units is $\{1, -1\}$ ; equivalently $u \mid 1$ holds exactly for $u = 1$ and $u = -1$ ).

Verification

technique · direct

1.1

Fact [L1] gives $\det A=2$ . Over $\mathbb Q$ , the matrix $\operatorname{diag}(1/2,1)$ multiplies with $A$ in either order to give $I_2$ , so $A$ is invertible there.

L1L2L3L4L5algebra

2.1

Over $\mathbb Z$ , the determinant $2$ is not a unit by [L5]. The contrapositive of [L2] therefore rules out an integer inverse.

step 1.1L2L5∎

Depends on

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: 83 results over 20 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.

Sources

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)

An integer matrix of determinant 222 is invertible over Q\mathbb QQ but not over Z\mathbb ZZ