TITLE: Decidability of a matrix product being the identity
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: There is a new algorithm (in manuscript) that shows that Identity Problem for SL(2,Z) is in NP, meaning that it is also NP-complete. 
Also point to point reachability for SL(2,Z) is also shown recently to be decidable, see proceeding of MFCS 2016 http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=6492
Which question was raised by Kontsevich and what is the connection with identity matrix? Could you please comment.