Around a Formula for the Rank of a Matrix Product with Some Statistical Applications

We consider the rank formula r(A*B) = r(A) + r(B) – r(A : B) + r(A*Q_BQ_AB), where Q_A and Q_B are the orthogonal projectors on the orthocomplements of the ranges, respectively, of A and B. We offer a short proof, and show how this formula leads to a strengthened version of Sylvester's Law of Nullity; we also obtain several extensions. These results are then applied to some problems in mathematical statistics concerned with the concepts of connectedness and orthogonality in the theory of experimental designs in the two-way and three-way layouts.