Composing and Inverting Transformations

A linear transformation moves every point in the plane by multiplying its coordinate vector by a matrix. If you apply transformation A first and then transformation B, you can find the combined effect by matrix multiplication: take a point, multiply by A to get the new point, then multiply that result by B. Because matrix multiplication is built to chain like this, doing B(Ax) for every point x is exactly the same as multiplying x by the single matrix BA. Notice A sits on the right, closest to x, because it acts first.

Order matters because the second transformation does not act on the original point, it acts on whatever the first transformation produced. Rotating a point and then stretching it sideways gives a different final location than stretching it first and then rotating it, since the stretch and the rotation each see a different starting shape to work on. That is why BA and AB are generally different matrices, even though both are built from the same two transformations.

This connects directly to inverses: if A takes a shape to a new position, A⁻¹ undoes exactly that move, and undoing a composed transformation means undoing the steps in reverse order, so (BA)⁻¹ = A⁻¹B⁻¹. Composing and inverting together let you build complicated motions, like rotating then scaling then reflecting, out of simple pieces, and then take any of it apart again step by step.