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.
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 below.