Given a finite set $S$ of $n\times n$ integer matrices, it is known that for $k\geq 3$ it is undecidable whether some product of them (allowing repetitions) is the zero matrix (called the mortality problem). It is also known (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.448&rep=rep1&type=pdf) that if $k=2$, then it is decidable whether some product of the matrices is the identity matrix. Is it known whether for $k$ sufficiently large, it is undecidable whether some product of matrices in $S$ is the identity matrix? This has some connection with a question raised by Kontsevich.
This is undecidable in dimension 4 or up see http://cgi.csc.liv.ac.uk/~igor/papers/matrixcomp.pdf
The result is proved in Bell, P. C., Potapov, I.: On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups, International Journal of Foundations of Computer Science, 21(6), 2010, 963–978.
The problem is at least NP-hard in dimension 2.