Denote by R n , m the ring of invariants of m-tuples of nxn matrices (m,n>=2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R n , m . We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))=<n, then R n , m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R 2 , m is determined for the case char(K)=2: it consists of 2 m +m-1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)<>2. We prove that the characterization of the R n , m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R 2 , 4 , and analyze the difference between the cases char(K)=2 and char(K)<>2.