If we define the inverse of a matrix A as another matrix A^{−1} such that

A^{−1}.A=I

then it is true, but not obvious, that also

A.A^{−1}=I.

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}

Which means

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].