  Let $(\xi_j)_{j\ge1} $, be a non-stationary Markov chain with phase space $X$
and let $\mathfrak{g}_j:\,X\mapsto\mathrm{SL}(m,\mathbb{R})$ be a sequence of
functions on $X$ with values in the unimodular group. Set
$g_j=\mathfrak{g}_j(\xi_j)$ and denote by $S_n=g_n\ldots g_1$, the product of
the matrices $g_j$. We provide sufficient conditions for exponential growth of
the norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. This
generalizes the classical theorem of Furstenberg on the exponential growth of
products of independent identically distributed matrices as well as its
extension by Virtser to products of stationary Markov-dependent matrices.
