The practical usage of Arnold Matrix Trace Theorem

Question

I would like to ask about the Arnold's Matrix Trace theorem: $$\textrm{tr}\big(A^{p^k}\big)\equiv\textrm{tr}\big(A^{p^{k-1}}\big)\ (\!\!\bmod {p^k}).$$ This theorem looks fantastic to me. But is there any practical usage of it or maybe some algorithm based on this?

EDITED: Adding some reference:

https://rjlipton.wordpress.com/2009/08/07/fermats-little-theorem-for-matrices/