For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(ω_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(ω):=\lim_{n\to \infty} \frac 1n \log \|A_{ω_0} A_{ω_2}\cdots A_{ω_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{ω_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(ω)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $ω$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $ω$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.