A matrix whose entries are all integers. Special cases which arise frequently are those having only as entries (e.g., Hadamard matrix), (0,1)-matrices having only as entries (e.g., adjacency matrix, Frobenius-König theorem, Gale-Ryser theorem, Hadamard's maximum determinant problem, hard square entropy constant, identity matrix, incidence matrix, Lam's problem), and those having as entries (e.g., alternating sign matrix, C-matrix).

The zero matrix could be considered a degenerate case of an integer matrix. Integer matrices are sometimes also called "integral matrices," although this usage should be deprecated due to its confusing use of the term "integral" as an adjective.