If a matrix A moves the plane, its inverse A^{-1} is the matrix that moves it right back. Apply one after the other and you return to exactly where you started — the identity:

The inverse undoes A the way \tfrac13 undoes 3. It exists only when A is invertible — that is, when \det A \neq 0. A transformation that collapses a dimension has thrown information away, and no matrix can bring it back.

Warp, then un-warp

Step through the stages. First the grid sits at rest. Apply A and it stretches and skews. Then apply A^{-1} and it snaps perfectly back to the original square grid — the round trip is the identity.

Why inverses run the world

Solving equations is inverting. To crack A\vec{x} = \vec{b} you multiply both sides by A^{-1} and read off \vec{x} = A^{-1}\vec{b}. Undoing a rotation, decoding a transformation, fitting a model — all are inverse problems. The catch is that A^{-1} only exists when nothing was crushed; the next page shows how to actually compute it for a 2×2.