Inverse Properties

Linear Algebra (Matrices) · Axiom Academy

Why inverting a product flips the order: (AB)^ -1 = B^ -1 A^ -1 — the same reason you take off your shoes before your socks. This morning you put on socks first, then shoes . Tonight, how do you take them off? You can't pull a sock through a shoe — you take the shoes off first, then the socks . The last thing you put on is the first thing you take off. Matrices behave exactly the same way. Apply B (socks), then A (shoes), and the combined action is the product AB . To undo it you must peel the actions off in the opposite order. Putting that into symbols gives the property at the heart of this lesson. The inverse of a product is the product of the inverses, in reverse order : undo the last action ( A ) first, then the first action ( B ) It is tempting to guess (AB)^ -1 = A^ -1 B^ -1 — same order, just inverted. The animation puts both candidates to the test on the same dressed-up grid: only the reversed order walks it back home. Each inverse meets its partner and collapses to the identity, from the inside out. A few more inverse identities round out the toolkit. Each one is just careful bookkeeping — nothing surprising once the reverse-order idea is in hand: Undo an undo and you are back where you started. Transpose and inverse commute — do either order. Scaling by k inverts to scaling by 1/k . The determinant of the inverse is the reciprocal.

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