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.

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

Example

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

Facts & Assumptions

Given: $A = (\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array})$ .

[L1]

The $2 \times 2$ determinant is $a d - b c$ (The Leibniz formula gives $\det (\begin{array}{cc} a & b \\ c & d \end{array}) = a d - b c$ ).

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

Verification

technique · direct

1.1

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

L1L2L3L4L5algebra

2.1

Over $Z$ , 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 · two levels

42 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

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

An integer matrix of determinant 22 is invertible over QQ but not over ZZ