There is a ring $R$ and its subring $K$ with unit. We have a matrix $A$ of order $n$ over $R$. Someone said, that if $A^m$ for $m=1,...,n$ can be represented as a sum of matrices over $R$ which a integral over $K$, than $A$ is integral over $K$. Why is that?



When powers of matrices are represented as a sum of integral matrices
A matrix is integral iff its eigenvalues are integral. Thus A is integral if and only if A^m is integral.


