Alphabeta Math

Determinants over a Ring: Examples and Counterexamples

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✓ 6 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Coxeter Groups
Linear Algebra
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

A full 3×3 Leibniz expansion lists all six permutations and their signs

Example

An integer matrix of determinant 2 is invertible over ℚ but not over ℤ

Counterexample

Over ℤ/2, an antisymmetric bilinear form need not be alternating

Counterexample

The naive quaternionic formula ad-bc is not row-alternating: a matrix with equal rows can have value 2k≠0

Example

Row operations track determinant correctly for a singular triangular matrix over ℤ/6

False statement

FALSE: det(A+B)=det(A)+det(B) for all same-sized square matrices

Sources