TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The determinant is the unique normalized alternating multilinear function on the columns

Statement

For $n\ge1$ over a commutative ring, determinant is the unique function $F:M_n(R)\to R$ that is column-multilinear, alternating and normalized.

Facts & Assumptions

Given: A normalized alternating column-multilinear function $F:M_n(R)\to R$ .

[L1]

Every alternating column-multilinear function satisfies the rigidity formula $F(A)=F(I_n)\det(A)$ (Every alternating multilinear $F$ satisfies $F(A)=F(I)\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_i a_{\sigma(i),i}$ ).

[L2]

The Leibniz determinant itself is alternating, column-multilinear and normalized (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring).

Proof

technique · direct

1.1

Fact [L2] proves existence of a function with the three properties.

L1L2

2.1

If $F$ has them, normalization gives $F(I_n)=1$ , so [L1] gives $F(A)=\det(A)$ for every $A$ . Hence the function is unique.

step 1.1L1algebra∎

Depends on

Used by

For same-sized finite square matrices over a commutative ring, det(AB)=det(A)det(B) Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 9 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 4.22 (standard reference, not scraped)
P. Massot, Structures algébriques fondamentales, Definition 6.4.1 (standard reference, not scraped)