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.

Row operations track determinant correctly for a singular triangular matrix over $Z / 6$

Example

Over $R = Z / 6$ , let $A = diag (2, 1)$ . Then $A$ is not invertible although $\det A = 2 \neq 0$ . A swap changes the determinant to $4 = - 2$ , scaling the first row by the nonzero nonunit $3$ changes it to $0 = 3 \cdot 2$ , and adding four times the second row to the first leaves it equal to $2$ .

Facts & Assumptions

Given: The displayed matrix over $Z / 6$ .

[L1]

Row swaps, arbitrary scalings and row additions have the three determinant laws without a nonsingularity assumption (For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged).

[L2]

A triangular determinant is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries).

[L3]

Arithmetic in $Z / 6$ is commutative ring arithmetic modulo $6$ (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L4]

$Z / 6$ consists of congruence classes modulo $6$ (The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[L5]

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

Verification

technique · direct

1.1

By [L2], $\det A = 2$ . The element $2$ is not a unit modulo $6$ , since every product $2 r$ is even and cannot be congruent to $1$ ; therefore [L5] implies that $A$ is not invertible.

L1L2L3L4L5algebra

2.1

Swapping rows gives $(\begin{array}{cc} 0 & 1 \\ 2 & 0 \end{array})$ with determinant $- 2 = 4$ . Scaling row $0$ by $3$ gives $diag (0, 1)$ with determinant $0 = 3 \cdot 2$ . Replacing row $0$ by row $0 + 4 R_{1}$ gives $(\begin{array}{cc} 2 & 4 \\ 0 & 1 \end{array})$ with determinant $2$ .

step 1.1algebra

3.1

These values agree respectively with negation, multiplication by $3$ , and invariance under row addition in [L1]. No cancellation, unit scaling or invertibility was used.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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)

Row operations track determinant correctly for a singular triangular matrix over Z/6Z/6