Matrix congruences

Question

If A, B are two integer square matrices of the same size such that $A\equiv B\pmod n$, is $A^p\equiv B^p \pmod{pn} $ for a prime p dividing n?

Answer 1

let $n=p=2$
let $A=\begin{pmatrix}0&0\\1&0\end{pmatrix}$ and $B=\begin{pmatrix}0&2\\1&0\end{pmatrix}$
$A^2$ is the zero matrix and $B^2=\begin{pmatrix}2&0\\0&2\end{pmatrix}$
$A\equiv B\pmod{2}$ but $A^2\not\equiv B^2\pmod{4}$