I am looking for a class of invertible matrices, $\{A_i\}$, where if I have a finite set of $n$ such matrices, $A_1,A_2,...,A_n$ that do not commute, then if I multiply them in some order to get a resulting matrix $S$, by looking at $S$ I can quickly "read off" the order that the $A_1,...,A_n$ were multiplied in to obtain $S$.
I am wondering if there are any obvious candidates for such a collection of matrices? My only idea is that the collection of matrices could be a collection of transposition matrices, so that when I give you the resulting permutation matrix obtained by multiplying the collection of transpositions in some way, then you can read off that ordering, but I am wondering if there is a better way, and especially a way where I can read off the ordering of the product "quickly".
What kind of matrices do you need? If you work over a field of characteristic zero (or anything that contains $\mathbb{Z}$), then you can choose a pair of matrices as here. They generate a free subgroup of $\mathrm{SL}_2(\mathbb{Z})$ under multiplication, and hence a free subsemigroup if we are to prohibit inverting them. For larger matrix sizes pad the diagonal with ones. Sorry for posting as a guest. Edit: proper reference (the linked question seems to consider $\mathbb{R}$).