  Given a finite irreducible set of real $d \times d$ matrices $A_1,\ldots,A_M$
and a real parameter $s>0$, there exists a unique shift-invariant equilibrium
state associated to $(A_1,\ldots,A_M,s)$. In this article we characterise the
ergodic properties of such equilibrium states in terms of the algebraic
properties of the semigroup generated by the associated matrices. We completely
characterise when the equilibrium state has zero entropy, when it gives
distinct Lyapunov exponents to the natural cocycle generated by
$A_1,\ldots,A_M$, and when it is a Bernoulli measure. We also give a general
sufficient condition for the equilibrium state to be mixing, and give an
example where the equilibrium state is ergodic but not totally ergodic.
Connections with a class of measures investigated by S. Kusuoka are explored in
an appendix.
