◄ WORLD V · SONNY 5DART 279 · a helldive at the net

THE DETERMINANT signed volume — and what a shear is forbidden to change

The determinant of a matrix is the signed volume of the box its columns span — and it is an invariant in two deep senses. It is unchanged by a shear (adding a multiple of one row to another leaves it fixed: a slanted box has the same volume as an upright one of equal base and height), and it is unchanged by a change of basis (similar matrices P⁻¹AP have equal determinants). That second invariance is why the determinant is a property of the linear map, not of the particular coordinates you wrote it in. It measures how a transformation scales volume — and the transformations that don't scale volume are exactly the ones it cannot see.

THE TECHNIQUE det unchanged by a shear (row-add) and by a change of basis

A 2×2 determinant, then the same matrix after a shear (row₁ += 2·row₂). The demo re-measures — the signed volume is unchanged: live demo







  
HISTORY & CREDIT Seki 1683 · Leibniz 1693 · Cauchy 1812

  
“Shearing a shape changes its size.” — a shear slides the top of the box sideways without changing its base or height, so the volume — the determinant — is exactly preserved. The determinant is built to be blind to shears. cited

  
1683 · Seki Takakazu and 1693 Leibniz — determinants, independently, for solving linear systems.
1812 · Cauchy — the systematic theory and the first use of determinant in its modern sense (the word itself Gauss coined, 1801, for a quadratic form's discriminant); the multiplicative law det(AB)=det(A)det(B) is the Cauchy–Binet formula — Binet proved it independently the same year.
now · the invariant that detects singularity, orientation, and volume scaling everywhere in linear algebra.

  
Adding one row to another slides the box but keeps its volume; changing coordinates renames the map but keeps its determinant. Two invariances, one number — a property of the map, not the grid. Cauchy 1812






  
RECOMMEND FOR I-13 determinant before and after a shear, computed

  
On the canonical compiler, det[[2,1],[1,3]] = 5; after the shear row₁ += 2·row₂ the matrix is [[4,7],[1,3]] and the determinant is 5 again — exactly:

  
$ i13 run inv_determinant.i13   # det, then det after a shear
det_before = 5
det_after  = 5
diff = 0            -- a shear cannot change the signed volume

  
Recommend: the determinant is invariance i13 computes exactly. Because the shear is an integer row operation, i13's f64 arithmetic stays exact (values well under 2⁵³), so the invariant holds with difference 0, not merely to precision. It is another foreign shape i13 enacts as a plain arithmetic program: the linear group's volume invariant, computed from the matrix entries. Where the cross-ratio (dart 278) carried the projective invariant, the determinant carries the linear one — and both show the pattern this batch is built on: transform the input, re-measure, and watch a deep quantity refuse to move.