  The theory of products of random matrices and Lyapunov exponents have been
widely studied and applied in the fields of biology, dynamical systems,
economics, engineering and statistical physics. We consider the product of an
i.i.d. sequence of $2\times 2$ random non-invertible matrices with real
entries. Given some mild moment assumptions we prove an explicit formula for
the Lyapunov exponent and prove a central limit theorem with an explicit
formula for the variance in terms of the entries of the matrices. We also give
examples where exact values for the Lyapunov exponent and variance are
computed. An important example where non-invertible matrices are essential is
the random Hill's equation, which has numerous physical applications, including
the astrophysical orbit problem.
