According to [this paper](http://arxiv.org/abs/1305.1965), a cube of a composition of matrix inverse, matrix elements' inverse, and matrix transposition modifies 3x3 matrix by multiplying on left and right side by diagonal matrices. 


A square of such composition is identity for 2x2 matrices over a commutative ring because in 2x2 case these operation commute and each squares to identity.


Therefore the questions: 


What are the intuitive reasons why the 3x3 case it true? 


Specifically, why ones need to cube a composition of order 2 elements to get something resembling an identity?



What is the intuition behind Kontsevich-Iyudu-Shkarin result?
The answer is contained in [Kontsevich's paper](http://arxiv.org/abs/1109.2469) (where he conjectures this result).


