This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller.

Let $\mathbb{N}$ denote the set of positive integers and for $n\in\mathbb{N}$ let $\text{Mat}(n\times n,\mathbb{Z})$ be the set of integer matrices of the format $n\times n$.

Denote with $B_n$ the set of matrices $A\in \text{Mat}(n\times n,\mathbb{Z})$ such that the sequence $(A^k)_{k\in\mathbb{N}}$ is eventually periodic; that is: $$B_n = \{A\in \text{Mat}(n\times n,\mathbb{Z}): \exists k, \ell\in\mathbb{N} (k\neq \ell \wedge A^k = A^\ell)\}.$$

Is $B_n$ computable for every $n\in\mathbb{N}$?

**Original version of the question**: Let $\mathbb{N}$ denote the set of positive integers and for $n\in\mathbb{N}$ let $\text{Mat}(n\times n,\mathbb{Z})$ be the set of integer matrices of the format $n\times n$.

For $A\in \text{Mat}(n\times n,\mathbb{Z})$ let the *most extreme entry* be defined by $$\text{m.e.}(A) = \max \big\{ |A_{ij}|:i, j\in\{1,\ldots,n\} \big\}.$$

We say $A\in \text{Mat}(n\times n,\mathbb{Z})$ is *bounded (with respect to the most extreme entry)* if there is $M\in\mathbb{N}$ such that for all $k\in\mathbb{N}$ we have $\text{m.e.}(A^k) \leq M$. By $B_n$ we denote the set of bounded matrices $A\in \text{Mat}(n\times n,\mathbb{Z})$ in the sense above.

Is $B_n$ computable for every $n\in\mathbb{N}$?