Visuri, Koivunen and Oja (2003) proposed and illustrated the use of the affine equivariant rank covariance matrix (RC,M) in classical multivariate inference problems. The RCM was shown to be asymptotically multinormal but explicit formulas for the limiting variances and covariances were not given. In this paper the influence functions and the limiting variances and covariances of the RCM and the corresponding scatter estimate are derived in the multivariate elliptical case. Limiting efficiencies axe given in the multivariate normal and t distribution cases. The estimates based on the RCM are highly efficient in the multinormal case, and for heavy-tailed distribution, perform better than those based on the regular covariance matrix. Finite-sample and asymptotic efficiency comparisons to a selected redecending M-estimator and S-estimator axe reported.