The mortality problem for $3\times 3$ matrices: given a finite set $F$ of $3\times 3$
integer matrices, decide whether the zero matrix is a product of members of $F$ 
(with repetitions allowed). This was proved unsolvable by Michael Paterson, 
Studies in Applied Mathematics 49 (1970), 105--107, doi: [10.1002/sapm1970491105](https://doi.org/10.1002/sapm1970491105).

The corresponding problem for $2\times 2$ matrices is apparently still open.

**Edit** (11 September 2016): The problem for $2\times 2$ matrices is apparently *still* open, despite the solution of a seemingly similar problem mentioned in Igor Potapov's [answer][1] below.


  [1]: https://mathoverflow.net/a/249542