Alphabeta Math

Determinants of Matrices over a Commutative Ring

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1 · Prerequisites
Abstract Algebra
Combinatorics
Coxeter Groups
Linear Algebra
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Finite rectangular matrices over a commutative ring, their entries, rows and columns

Definition

Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose

Theorem

Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products

Definition

Invertible square matrices and similarity over a commutative ring

Proposition

For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface

Definition

Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring

Lemma

Every alternating multilinear matrix function is antisymmetric under a column swap

Lemma

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

Definition

For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix

Corollary

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

Theorem

The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring

Theorem

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

Corollary

A square matrix with a zero column or two equal columns has determinant zero

Theorem

For every square matrix over a commutative ring, det(A^T)=det(A)

Corollary

The determinant is alternating and multilinear in the rows as well as in the columns

Definition

Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished

Theorem

For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged

Definition

Upper triangular, lower triangular and diagonal square matrices over a commutative ring

Theorem

The determinant of a triangular matrix is the product of its diagonal entries

Theorem

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

Corollary

An invertible square matrix over a commutative ring has unit determinant

Theorem

A finite square real matrix is invertible if and only if its determinant is nonzero

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

Example

The Leibniz formula gives detbeginpmatrixa&bc&d endpmatrix=ad-bc

Sources