# Determinant

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/determinant/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A single number from a square matrix revealing invertibility and area/volume scaling factor. Zero determinant means the transformation collapses dimensions.

## 2x2 formula
det[[a,b],[c,d]] = ad−bc

## Key properties
det(I)=1; det(AB)=det(A)det(B); det(A⁻¹)=1/det(A); row swap flips sign; proportional rows → det=0

## Determinant = product of eigenvalues

## Historical origins (before matrices!)
Seki Takakazu (Japan, 1683), Leibniz (1693) — predate Cayley's matrix theory by ~170 years. Term coined by Gauss (1801); modern sense and multiplicative property by Cauchy (1812).

## Leibniz formula (general n×n)
det(A) = Σ sgn(σ)·a₁,σ(1)···aₙ,σ(n) — sum over all n! permutations

## Jacobian determinant
Scaling factor for multivariable change-of-variables in integration.

## Sources
### Tier 1
- Strang, G. (2016). *Introduction to Linear Algebra*. 5th ed.
- Axler, S. (2015). *Linear Algebra Done Right*. 3rd ed.
### Tier 2
- Kline, M. (1972). *Mathematical Thought from Ancient to Modern Times*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*