We explain the necessary and sufficient conditions for recurrent and transi
ent behavior of a random walk in a stationary ergodic random environment on
a strip in terms of properties of a top Lyapunov exponent. This Lyapunov e
xponent is defined for a product of a stationary sequence of positive matri
ces. In the one-dimensional case this approach allows us to treat wider cla
sses of random walks than before.