  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 $3\times 3$ 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
$(J_2\circ J_1)^3$ is the identity map modulo the ${\rm Diag}_{L} \times
\rm{Diag}_R$ 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 $(J_2\circ J_1)^3$ is
defined, there are invertible diagonal $3\times 3$ matrices $D_1=D_1(M)$ and
$D_2=D_2(M)$ such that $(J_2\circ J_1)^3(M)=D_1^{-1}MD_2.$
