Let $\Gamma_A$ denote the abelian-by-cyclic group associated to an
integer-valued, non-singular matrix $A$. We show that if $A$ has no eigenvalues
of modulus one, then there are no faithful $C^1$ perturbations of the trivial
action $ \iota: \Gamma_A \to \diff$, where $M$ is a compact manifold.