A theorem giving attainable upper and lower limits for the trace of certain products of real matrices is established. These products are of the form X1r1X2r2...Xnrn with orthogonal matrices Xi and diagonal matrices ri where the matrices Xi are allowed to vary independently and unrestrictedly. The proof makes use of two lemmas. The theorem may find application in psychometrics when the trace of matrices is involved. Several examples taken from this area are appended by way of illustration.