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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 offunctions 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 ofthe matrices $g_j$. We provide sufficient conditions for exponential growth ofthe norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. Thisgeneralizes the classical theorem of Furstenberg on the exponential growth ofproducts of independent identically distributed matrices as well as itsextension by Virtser to products of stationary Markov-dependent matrices.

Exponential growth of products of non-stationary Markov-dependent matrices