Undoing Matrix Operations

Linear Algebra (Matrices) · Axiom Academy

Every reversible matrix has a twin that runs it backward: apply A , then apply A^ -1 , and every point lands right back home — A^ -1 A = I . A move, and the move that takes it back A matrix is an action on space — it can rotate, shear, stretch, or skew every point at once. The big idea of this unit is that most of those actions can be reversed . The matrix that runs A backward is its inverse , written A^ -1 : do A , then do A^ -1 , and the two cancel. You don't have to take that on faith — watch it happen, then drive it yourself. Watch the figure get skewed by , then watch pull it back. It doesn't land near the start — it snaps to exactly the start, because A^ -1 A = I , the do-nothing matrix. The dashed outline never moves — it marks where the figure started. The round trip A^ -1 A brings it home, dead on. Press Apply A to skew the figure, then Apply A^ -1 (undo) to run it backward. Watch the position readout: it only reports home when you're exactly back where you started — that's the inverse cancelling A perfectly. Not every action can be reversed. Switch between an invertible matrix and a singular one ( ). Apply it, then try to undo. The invertible one returns home; the singular one flattens the whole figure onto a single line — and once two different points land on the same spot, no matrix can tell them apart again. The undo button greys out: A^ -1 does not exist .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.