In this paper the nilpotency properties of deadbeat linear multivariable discrete-time disturbance-rejection systems incorporating full-order or reduced-order observers for plants with accessible or inaccessible statea are investigated. The theory is illustrated by the presentation of the results of computer simulation studies which indicate that the presence of an observer affects only the transient behaviour of deadbeat discrete-time disturbance-rejection systems and that the use of a reduced-order observer results in markedly superior transient behaviour than the use of a full-order observer.