LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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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.
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Every alternating multilinear matrix function is antisymmetric under a column swap

Statement

Let $n \geq 1$ and let $F : M_{n} (R) \to R$ be column-multilinear and alternating. Interchanging any two columns of $A$ negates $F (A)$ . This holds over every commutative ring, including characteristic $2$ .

Facts & Assumptions

Given: An alternating column-multilinear function $F$ and two selected column positions containing $u$ and $v$ .

[L1]

Alternation makes $F$ vanish on equal columns, while multilinearity expands sums separately in each column (Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring).

[L2]

Ring addition has additive inverses and satisfies the usual sign identities (In any ring $0 \cdot a = a \cdot 0 = 0$ , $(- a) b = a (- b) = - (a b)$ , $(- a) (- b) = a b$ , $(- 1) a = - a$ and $a (b - c) = a b - a c$ ).

Proof

technique · direct

1.1

Put $u + v$ in both selected positions. Alternation gives $0 = F (\dots, u + v, \dots, u + v, \dots)$ .

2.1

Expanding twice gives $0 = F (\dots, u, \dots, v, \dots) + F (\dots, v, \dots, u, \dots)$ because the two equal-column terms vanish. Adding the inverse of the first term yields the antisymmetry formula, without dividing by $2$ .

step 1.1L1L2algebra∎

Depends on

Used by

Every alternating multilinear F satisfies F(A)=F(I)∑_σ∈ Sₙsgn(σ)∏ᵢ a_σ(i),i Lemma

Dependency tree · two levels

10 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, Theorem 4.19 (standard reference, not scraped)