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For each nonnegative integer $k$, you could find the possible tuples of matrices $(A_1, ..., A_k)$ as follows:

• for each of -1, 0, 1, find the ways to write it as a sum of $k$ summands each in {-1, 0, 1}.

• the target matrix $M$ has $n^2$ entries $m_{i, j}$; so the $(A_1, ..., A_k)$ are obtained by combining all the possible ways to get each entry.