NapkinCalc

Linear Algebra B — Inverses & Solving Systems

The inverse — undoing the machine

ELI5: M⁻¹ reverses whatever M did, just like 1/x undoes ×x. It exists only when det ≠ 0 — a machine that flattened space (det = 0) destroyed information and can't be reversed. The defining property is M·M⁻¹ = the identity, and a clean consequence is det(M)·det(M⁻¹) = 1.

M:=[2,1;1,3]M := [2, 1; 1, 3] = [[2,1],[1,3]][[2, 1], [1, 3]] the machine
Minv=inv(M)M_{inv} = \mathrm{inv}\left(M\right) = [[0.6,−0.2],[−0.2,0.4]][[0.6, -0.2], [-0.2, 0.4]] its undo machine
dinv=det(Minv)d_{inv} = \mathrm{det}\left(M_{inv}\right) = 0.20000.2000 det 1/5 — undoing a 5× stretch
✓ pass abs(det(M)⋅dinv−1)<10−9\mathrm{abs}\left(\mathrm{det}\left(M\right) \cdot d_{inv} - 1\right) < 10^{-9} det(M)·det(M⁻¹) = 1, always

Real-world hook: inverses decode what a transform encoded — un-rotating a scan, deconvolving a blur, solving "what inputs produced these outputs?" When det = 0, that information is simply gone.