Let $\xi_1,\xi_2,\dots$ be a stationary ergodic Markovian process on a measurable space $\Xi$ and $X$ be a measurable mapping of $\Xi$ into the group $SL(m,R)$. We prove that, under some conditions, the norm of the product $$ X(\xi_1)X(\xi_2)\dots X(\xi_n) $$ of random unimodular matrices grows exponentially with probability 1 (Theorem 1). The proof is based on some facts from the theory of unitary representations of the group $SL(m,R)$ and on the theorem on the exponential decrease of the mean of the product of random unitary operators on a separable Hilbert space (Theorem 2).