A simple decision problem whose decidability is not known
The Matrix Mortality Problem for 2x2 matrices.  I.e., given a finite list of 2x2 integer matrices M1,...,Mk, can the Mi's be multiplied in any order (with arbitrarily many repetitions) to produce the all-0 matrix?

(The 3x3 case is known to be undecidable.  The 1x1 case, of course, is decidable.)