If we define the inverse of a matrix A as another matrix A−1 such that
then it is true, but not obvious, that also
Remember: I is the identity matrix, the matrix which corresponds to the transformation which is just leaving things exactly as they are.
If A−1.A=I, then multiply each side before by A and after by A−1.
A.A−1.A.A−1 = A.A−1
If A−1 exists, then det A ≠ 0, so det A−1 ≠ 0 too, and det A.A−1 ≠ 0. So A.A−1 has an inverse. Call it B. Then:
B.A.A−1.A.A−1 = B.A.A−1
A.A−1 = I ▇
In other words, the general truth that with matrices M.N may not be equal to N.M doesn’t apply with inverses. A.A−1 is the same as A−1.A. Just as well, because otherwise you would have to calculate two inverses for each matrix, one for multiplying before, and another for multiplying after.
Another fact about inverses:
If A and B have inverses, then:
(AB)−1 = B−1.A−1
You can see why by thinking about socks and shoes.
The inverse of [put on shoes].[put on socks].[feet]
is [take off socks].[take off shoes].[feet].