We investigate three fundamental equivalences in the representation theory of algebras and groups: Morita, derived and stable equivalences. Mainly we focus on them for the class of the centralizer algebras of matrices. This class of algebras appears in many aspects of mathematics. For example, in geometry variety, Markov process, and invariant theory. By introducing new equivalence relations on square matrices, we describe Morita, derived and stable equivalences of Morita type between centralizer algebras in terms of elementary divisors of given matrices. The talk reports parts of a joint work with X. G. Li.