Equations · equation
Determinant
Vectors & Geometry · 1683
- Equation
- det [a b; c d] = ad − bc; det of 3×3 = a(ei − fh) − b(di − fg) + c(dh − eg)
- Field
- Vectors & Geometry
- Symbols
- a…i the entries of a square matrix · det a single number: the signed area (2×2) or volume (3×3) spanned by the rows
- Year
- 1683
Significance
The scale factor of a linear map — how much a transformation stretches area or volume, with a sign for whether it flips orientation. Zero means the map squashes space flat and cannot be undone. The cross product is a 3×3 determinant; the Jacobian in every change of variables is one; the volume of a crystal's unit cell is the determinant of its three lattice vectors. Seki Takakazu had it in 1683, Leibniz independently a decade later.
Open on the Equations shelf — the interactive card, the filters, and everything alongside it.