Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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The Leibniz formula gives det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc

Example

For every commutative ring, det⁡(abcd)=ad−bc.\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc.

Facts & Assumptions

Given: A 2×22\times2 matrix A=(abcd)A=\begin{pmatrix}a&b\\c&d\end{pmatrix} over a commutative ring.

[L1]

Determinant is the signed Leibniz sum over S2S_2 (For n≥1n\ge1, the determinant over a commutative ring by the Leibniz formula, and ∣det⁡A∣|\det A| for a real matrix).

[L2]

Sign is (−1)(-1) to the inversion number (Inversions, inversion number, the sign sgn⁡(σ)=(−1)inv⁡(σ)\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}, and even and odd permutations).

Verification

technique · direct
1.1

The two permutations of 2={0,1}2=\{0,1\} are the identity, with sign 11, and the swap, with one inversion and sign −1-1.

L1L2
2.1

Their Leibniz monomials are a00a11=ada_{00}a_{11}=ad and a10a01=cb=bca_{10}a_{01}=cb=bc, so the signed sum is ad−bcad-bc.

step 1.1algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources