  The problems that we consider in this paper are as follows. Let A and B be
2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating
w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the
largest (by the absolute value) possible entry of W, over all w(A, B) of length
n, as a function of n? What is the expected absolute value of the largest (by
the absolute value) entry in a random product of n matrices, where each matrix
is A or B with probability 0.5? What is the Lyapunov exponent for a random
matrix product like that? We give partial answer to the first of these
questions and an essentially complete answer to the second question. For the
third question (the most difficult of the three), we offer a very simple method
to produce an upper bound on the Lyapunov exponent in the case where all
entries of the matrices A and B are nonnegative.
