We consider iterated function systems that contain inverses in the overlapping case. Under the transversality condition, we show that the invariant measure is absolutely continuous for a.e. parameter when the random walk entropy is greater than the Lyapunov exponent. We also show that if the random walk entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value.