Strang §2.5 · Inverse of a product
Undo in reverse order.
AB means B first, then A. To undo it, you undo the last move first: A⁻¹ first, then B⁻¹. Written as a product, that is B⁻¹A⁻¹.
(AB)⁻¹ = B⁻¹A⁻¹ (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹
Think: socks and shoes, taken off
You put on socks, then shoes. In the evening you take off the shoes first, then the socks. Same two steps, reverse order.
Matrix products read right to left. AB puts on B first. So the inverse takes off A first, and that is written B⁻¹A⁻¹.
Check it with algebra
(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹ I B = B⁻¹B = I ✓
The A and its inverse meet in the middle and cancel. Then B and its inverse cancel.
Worked example
R turns by 90°, S is a shear. From lesson 1, RS = 0−111
R⁻¹ = 01−10 (turn back) S⁻¹ = 1−101 (slide back)
S⁻¹R⁻¹ = 11−10 but R⁻¹S⁻¹ = 01−11
Only the first one undoes RS.
Where you meet this
A robot arm's pose is a product of joint matrices: T₁T₂⋯Tₙ takes a point from the gripper's frame to the base's frame. To go the other way, from the base to the gripper, you need the inverse, Tₙ⁻¹⋯T₁⁻¹. The order of the joints reverses.
A rotation is undone by turning back, and that matrix is also its transpose. Slide 5 explains why that matters.