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.

Give a specific $k>0$, I am getting trouble by finding its closed-form.

Hints and comments are welcomed. Thanks.

EDIT:

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

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

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

STEP 3: Combine the results from STEP2.