Let M n (Z) be the ring of n-by-n matrices with integral entries, and n 2. This paper studies the set G n (Z) of pairs (A, B) ∈ M n (Z) 2 generating M n (Z) as a ring. We use several presentations of M n (Z) with generators X = n i=1 E i+1,i and Y = E 11 to obtain the following consequences.(1) Let k 1. The following rings have presentations with 2 generators and finitely many relations: (a) k j =1 M m j (Q) for any m 1 , . . . , m k 2. (b) k j =1 M n j (Z), where n 1 , . . . , n k 2, and the same n i is repeated no more than three times.(2) Let D be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for X and Y to describe all representations of the ring M n (D) into M m (D) for m n.(3) We obtain information about the asymptotic density of G n (F ) in M n (F ) 2 over different fields, and over the integers.