COMMENT ON EXERCISE 5.4.13

As Richard remarks, his method would be impractical for matrices much
larger than 3 x 3.  In fact, the intended method is much simpler (though
more abstract):  Look at the matrices as representing linear operators,
and ask how the rank of AB is related to the ranks of A and B.  For
example, if B has a nontrivial nullspace, what does that tell you about
AB?  The rest is up to you.  It may help to remember Exercise 4.1.6.

What Richard did is to prove Cramer's Rule for 3 x 3 matrices, and show
that the result is both a left and a right inverse.  I don't think that
this completely answers the original question.  What if det B (the
denominators) equals zero?  Is it clear that the equation AB = I can't
ever hold then?  (We know that (det A)(det B) = det(AB) = 1, but that is a
more advanced theorem, not proved in the textbook, so using it here would
amount to a circular argument.)