  We consider one-step cocycles of $2 \times 2$ matrices, and we are interested
in their Lyapunov-optimizing measures, i.e., invariant probability measures
that maximize or minimize a Lyapunov exponent. If the cocycle is dominated,
that is, the two Lyapunov exponents are uniformly separated along all orbits,
then Lyapunov-optimizing measures always exist, and are characterized by their
support. Under an additional hypothesis of nonoverlapping between the cones
that characterize domination, we prove that the Lyapunov-optimizing measures
have zero entropy. This conclusion certainly fails without the domination
assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles;
indeed we show that in the latter case there are measures of positive entropy
with zero Lyapunov exponent.
