1. (i) If m < p , the matrix S, and hence the matrix S/m (which is an unbiased estimate of the unknown covariance matrix of the underlying p-variate distribution), is singular. If m p, it is nonsingular with probability 1. (ii) If r + m 5 p, the test q,(y, u, z) E IX is the only test that is invariant under the groups G, and G3 of Section