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For an arbitrary associative unital ring $R$, let $J_1$ and $J_2$ be thefollowing 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 matrixinverse) and $J_2(M)_{jk}=(M_{kj})^{-1}\,$ (the transpose of the Hadamardinverse).  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 diagonalmatrices.  That is, we show that for each $M$ in the domain where $(J_2\circ J_1)^3$ isdefined, there are invertible diagonal $3\times 3$ matrices $D_1=D_1(M)$ and$D_2=D_2(M)$ such that $(J_2\circ J_1)^3(M)=D_1^{-1}MD_2.$

The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations