Undecidability of the stabilizer and zero-in-the-corner problems for matrix groups

The upper-left-corner problem and upper-right-corner problem for matrix semigroups with integer entries are known to be undecidable. Here we study the analogous problems for matrix groups and prove their undecidability for rational entries, by reducing to the undecidability of the word problem for groups. To reach this goal, we answer a question of Dixon from 1985 by proving the undecidability of the stabilizer problem for matrix groups. Our results can also be interpreted as showing the undecidability of the multi-matrices analogue of the classical Skolem problem.