  In this work, we give a rigorous explicit formula for the Lyapunov exponent
for some binary infinite products of random $2\times 2$ real matrices. All
these products are constructed using only two types of matrices, $A$ and $B$,
which are chosen according to a stochastic process. The matrix $A$ is singular,
namely its determinant is zero. This formula is derived by using a particular
decomposition for the matrix $B$, which allows us to write the Lyapunov
exponent as a sum of convergent series. Finally, we show with an example that
the Lyapunov exponent is a discontinuous function of the given parameter.
