DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix

Definition

Let $R$ be a commutative ring, let $n \geq 1$ , and let $A = (a_{i j}) \in M_{n} (R)$ . Its determinant is $\det (A) : = \sum_{σ \in S_{n}} sgn (σ) \prod_{i < n} a_{σ (i), i} .$ The sum is finite because $S_{n}$ has $n!$ elements. The product is taken in $R$ , so the signs $1$ and $- 1$ act through the ring identities. The formula uses columns indexed by $i < n$ and rows indexed by $σ (i)$ .

When $A$ is real, $∣ \det A ∣$ means the ordinary real absolute value of its real determinant. It is not alternate notation for the determinant itself.

For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix

Definition

Depends on

Dependency tree · 2 levels

Sources

For n≥1n≥1, the determinant over a commutative ring by the Leibniz formula, and ∣det⁡A∣∣detA∣ for a real matrix

Definition

Depends on

Dependency tree · 2 levels

Sources

For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix