On Wed, Nov 13, 2013 at 1:48 PM, Warren D Smith <warren.wds@gmail.com>wrote:

...

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.

I don't think so, because of the "up to a bit of messing around". In the commutative case, the "up to a bit of messing around" is "up to multiplication by some diagonal matrix. If the entries in this diagonal matrix were themselves polynomial or rational in the original 9 variables, then the Schwartz-Zippel lemma would apply, and the theorem would also be provable by any decent computer algebra system. But Schwartz-Zippel only applies to "are these two polynomials identical?", not to "are these polynomials related by a constant multiple?" Andy

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