Right alternative unital bimodules over the matrix algebras of order $\geq 3$

We address the unital right alternative bimodules over the matrix algebras $\mathrm{M}_n(\Phi)$ of order $n\ge3$, prove that each of these bimodules is the direct sum of an associative bimodule and a Graves bimodule, and fully describe the structure of twisted Graves bimodules. Also, we construct an irreducible right alternative $\mathrm{M}_n(\Phi)$-bimodule of minimal dimension $n(n-1)$. Furthermore, we show that no element $f(x,y)$ of the free right alternative algebra of rank 3 is its nuclear element. The results of this article are needed for the study of the right alternative superalgebras whose even part includes $\mathrm{M}_n(\Phi)$ with $n\ge3$.