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.

翻译：本文研究如下问题：设A和B为（实数域上的）2x2矩阵。令w(A,B)为长度为n的词，将w(A,B)作为矩阵乘积求值后得到2x2矩阵，记为W。作为n的函数，在所有长度为n的词w(A,B)中，W的（按绝对值）最大可能元素是多少？在由n个矩阵（每个矩阵以0.5概率取A或B）的随机乘积中，其（按绝对值）最大元素的期望绝对值是多少？此类随机矩阵乘积的Lyapunov指数为何？我们对第一个问题给出部分解答，对第二个问题给出本质上完整的解答。针对第三个问题（三者中最困难的），我们提出一种非常简洁的方法，在矩阵A和B所有元素均为非负的情形下给出Lyapunov指数的上界。