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

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

Statement

For $n \geq 1$ 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 $F (A) = F (I) \sum_{σ \in S_{n}} sgn (σ) \prod_{i} a_{σ (i), i}$ ).

[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

Dependency tree · two levels

13 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 4.22 (standard reference, not scraped)
P. Massot, Structures algébriques fondamentales, Definition 6.4.1 (standard reference, not scraped)