The inverse undoes A.
A⁻¹ is the matrix that undoes A. Apply A, then A⁻¹, and every point is back where it started.
Think: rewind
A moves the graph paper: it stretches, turns or slants it. A⁻¹ plays that film backwards. After both, nothing has moved.
"Nothing has moved" is the identity matrix I, with 1s on the diagonal and 0s elsewhere. Ix = x for every x.
Why it matters
If A has an inverse, Ax = b has exactly one solution for every b: x = A⁻¹b. Multiply both sides of Ax = b by A⁻¹ and the A disappears.
Worked example: undo a shear
Not every matrix can be undone
Think of the boat from lesson 1 with both thrusters pointing forward, thruster 2 pushing twice as hard. Throttles (2, 0) and (0, 1) move the boat exactly the same way. Watching the boat, you cannot tell which settings produced the motion.
That is a matrix with no inverse. It sends two different inputs to the same output, so there is no way back. Four equivalent tests, all from earlier slides:
- A has n nonzero pivots.
- The columns of A are independent.
- det A ≠ 0: the plane is not squashed.
- Ax = 0 only for x = 0.