NapkinCalc

Linear Algebra — Matrices

The inverse — undoing the machine

continues from lesson 1 — values defined earlier in the course stay live here

M⁻¹ undoes M, exactly like 1/x undoes multiplication. It exists precisely when det ≠ 0. The check verifies the defining property on an entry of M·M⁻¹ = I:

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