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

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For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries

Statement

For each fixed $n \geq 1$ , the function $\det : M_{n} (R) \to R$ is evaluation of the polynomial $P_{n} : = \sum_{σ \in S_{n}} sgn (σ) \prod_{i < n} x_{σ (i), i}$ in the $n^{2}$ commuting variables $x_{r i}$ .

Facts & Assumptions

Given: A fixed natural $n \geq 1$ and commuting indeterminates $x_{r i}$ for $r, i < n$ .

[L1]

The determinant is the finite Leibniz sum $\sum_{σ} sgn (σ) \prod_{i} a_{σ (i), i}$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L2]

Iteration constructs a polynomial ring in any finite list of commuting indeterminates over a commutative ring (Polynomial rings in finitely many commuting indeterminates by iteration).

[L3]

The real numbers form a field (The reals form a field).

[L4]

Every field is a commutative ring with the same operations (Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring).

[L5]

Evaluation at a ring element is a unital ring homomorphism, so it respects finite sums and products (Universal property of $R [x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism).

Proof

technique · direct

1.1L2L3L4

By [L2]–[L4], the displayed finite sum of signed monomials is an element $P_{n}$ of the real polynomial ring in the $n^{2}$ variables $x_{r i}$ .

2.1step 1.1L1L2L5algebra

Iterate [L5] through the finite list of variables, sending each $x_{r i}$ to $a_{r i}$ . The resulting homomorphism sends each monomial to $\prod_{i < n} a_{σ (i), i}$ and therefore sends $P_{n}$ to the Leibniz sum $\det (A)$ from [L1].

3.1step 2.1∎

Hence determinant is the polynomial function given by this explicit finite sum of finite products in the matrix entries.

Depends on

Dependency tree · 2 levels

Sources

P. Massot, Structures algébriques fondamentales, §6.4