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This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A , B] = 0 for two matrices A and B if and only if a vector v (B) defined uniquely from the matrix B is in the null space of a well- structured matrix defined as the Kronecker sum A ⊕ (- A*), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning commutators of certain matrices with functions of matrices f (A) which extends the well- known sufficiency -- type commuting result [A , f (A)] = 0. [ABSTRACT FROM AUTHOR]