Given $m$ $n\times n$ matrices $A_{1},A_{2} \dots A_{m}$ and a matrix $B$, is there a way to determine whether there's permutation $p$ such that for matrices $A_{i}$, $A_{p_{1}}A_{p_{2}} \dots A_{p_{m}} = B$. The solution should be "almost correct" in all cases, and hopefully, in polynomial time.
Moreover, one way is to check whether $\prod{\det(A_{i})} = \det(B)$, however it is "correct on almost every cases". I tried to find some other values to estimate its possibility but failed. Are there any possible solutions?
EDIT: Almost correct here means for $\epsilon$ possibility being wrong, you can get it in $O(f(\epsilon))$ time for some $f$, still we want $f$ not being to large. And some one pointed out that it may be a NP problem in the comment.
