  The Tits alternative states that a finitely generated matrix group either
contains a nonabelian free subgroup $F_2$, or it is virtually solvable. This
paper considers two decision problems in virtually solvable matrix groups: the
Identity Problem (does a given finitely generated subsemigroup contain the
identity matrix?), and the Group Problem (is a given finitely generated
subsemigroup a group?). We show that both problems are decidable in virtually
solvable matrix groups over the field of algebraic numbers
$\overline{\mathbb{Q}}$. Our proof also extends the decidability result for
nilpotent groups by Bodart, Ciobanu, Metcalfe and Shaffrir, and the
decidability result for metabelian groups by Dong (STOC'24). Since the Identity
Problem and the Group Problem are known to be undecidable in matrix groups
containing $F_2 \times F_2$, our result significantly reduces the decidability
gap for both decision problems.
