For an arbitrary associative unital ring R , let J 1 and J 2 be the following noncommutative birational partly defined involutions on the set M 3 (R) of 3×3 matrices over R : J 1 (M)=M −1 (the usual matrix inverse) and J 2 (M) jk =(M kj ) −1 (the transpose of the Hadamard inverse).
We prove the following surprising conjecture by Kontsevich saying that (J 2 ∘J 1 ) 3 is the identity map modulo the Diag L ×Diag R action (D 1 ,D 2 )(M)=D −1 1 MD 2 of pairs of invertible diagonal matrices.
That is, we show that for each M in the domain where (J 2 ∘J 1 ) 3 is defined, there are invertible diagonal 3×3 matrices D 1 =D 1 (M) and D 2 =D 2 (M) such that (J 2 ∘J 1 ) 3 (M)=D −1 1 MD 2 .