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 same-sized finite square matrices over a commutative ring,
Statement
For and over a commutative ring,
Facts & Assumptions
Given: A fixed matrix and a variable matrix .
Determinant is the unique normalized alternating column-multilinear matrix function (The determinant is the unique normalized alternating multilinear function on the columns).
Every alternating column-multilinear satisfies (Every alternating multilinear satisfies ).
Matrix multiplication is associative, distributive and unital (Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products).
Determinant is defined for every square matrix over the commutative ring (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix).
Proof
Define . Each column of is times the corresponding column of , so distributivity makes column-multilinear; equal columns of give equal columns of , so is alternating.
Since , one has .
Apply [L2] to : . Neither matrix was assumed invertible.
Depends on
- The determinant is the unique normalized alternating multilinear function on the columns
- Every alternating multilinear $F$ satisfies $F(A)=F(I)\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_i a_{\sigma(i),i}$
- Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
Used by
- An invertible square matrix over a commutative ring has unit determinant Corollary
- If A is invertible over a commutative ring, then det(A⁻¹)=det(A)⁻¹ Corollary
- Similar matrices over a commutative ring have the same determinant Corollary
- For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
- The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
- Change of variables for an injective C¹ map on a compact Jordan set Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 67 results over 14 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, Theorem 4.24 (standard reference, not scraped)
- P. Massot, Structures algébriques fondamentales, Lemma 6.4.2 (standard reference, not scraped)