[math-fun] Mike Stay's goofy identity

The conjecture was recently proved: http://golem.ph.utexas.edu/category/2013/10/who_ordered_that.html

Start with a 3?3 matrix. Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix. Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse. Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse. Theorem: Up to a bit of messing about, you?re back where you started.

--I have no idea what motivated this, but it is trivial to prove if by "prove" we mean, you are willing to accept probabilistic proof. I.e. using the Schwartz-Zippel lemma.