Summary: The families of \(V\)-variable fractals for \(V = 1,2,3, \ldots\), together with their natural probability distributions, interpolate between the corresponding families of random homogeneous fractals and of random recursive fractals. We investigate certain random \(V \times V\) matrices associated with these fractals and use them to compute the almost sure Hausdorff dimension of \(V\)-variable fractals satisfying the uniform open set condition.