Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix} 0 & 1\\ i^k & 1 \end{bmatrix}?$$

I know if $k=0$, we can use the matrix diagonalization method.

However, if $k>0$, it seems that the matrix diagonalization method is not suitable for those cases.

Given a specific $k>0$, I am having trouble finding its closed-form.

Hints and comments are welcomed.

EDIT:

An experiment for computing $\prod\limits_{i=1}^{n} A_i$ based on number theory:

STEP 1: Choose some different prime numbers $p_1,p_2,\dotsc$.

STEP 2: Compute $\prod\limits_{i=1}^{n} A_i \bmod p_1$, $\prod\limits_{i=1}^{n} A_i\bmod p_2,\dotsc$.

STEP 3: Combine the results from STEP 2.