Let $x \in \mathbb{R}^n$. Consider an $n\times n$ matrix $A$. Suppose we're interested in how $||A^nx||$ grows with $n$, the answer (excluding pathological cases) is that it scales exponentially with the exponent being $\log |\lambda(A)|$ where $\lambda(A)$ is the eigenvalue of $A$ with the largest magnitude.
Now suppose we have two matrices $A$ and $B$, and a biased coin $\sim$ Bernoulli($p$). For each time instant $i \geq 1$, we toss the coin and take $$x_{i+1} = A*x_i, ~~~~~w.p.~~~~~ p,$$ $$x_{i+1} = B*x_i, ~~~~~w.p.~~~~~ 1-p.$$ Let $x_0$ be some fixed vector. This is like a multiplicative random walk, where the random element is a matrix instead of a scalar. The question we ask is:
How does the $\mathbb{E}[||x_n||]$ scale with $n$? Is it exponential, and if so what is the exponent? For special cases (say $A$ and $B$ commute, or that they are orthogonal) the answer is easy to get. What can we say about general cases? An upper bound which holds is $$p\log|\lambda(A)| + (1-p)\log|\lambda(B)|.$$ Do we have any other bounds?