The results of this note yield the mutual independence of certain matrices, characteristic roots, Hotelling's $T^2$ or Mahalanobis' $D^2$ statistics, and C. R. Rao's $R$ statistic. The result concerning the mutual independence of certain of Hotelling's $T^2$ statistics has been proved by K. S. Rao [1]. The results mentioned here occur (by implication and as a by-product) in [5], [6] and [7] in the course of investigations on some specific problems in statistical inference but are not explicitly stated. These results can be utilised in statistical inference and especially in simultaneous tests and simultaneous confidence interval estimation, and also in other problems.