Imagine you have a $n\times n$ matrix filled in with permutations over $n$ elements. Now you pick one permutation from each row randomly starting from the first row and by multiplying them get a permutation $P_1$. You repeat this until you get $l$ distinct permutations. Now you want to recover the matrix (or at least some of its elements) from $P_1,...,P_l$.

What should be $l$ to make it theoretically possible? How computationally hard would be to recover the matrix?