  This paper is concerned with random walks on a family of dyadic-valued
solvable matrix groups. A description of the Poisson boundary of these groups
for probability measures of finite first moment and non-zero displacements (or
drifts) is given. When non-trivial, the boundary may be identified with a space
of matrices with real and 2-adic entries, depending on the values in a
displacement matrix associated with the random walk. Conditions for boundary
triviality are also discussed.
