# Decidability of a matrix product being the identity

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.

-