Commuting pairs of bounded 3x3 integer matrices are shown to number Theta(N^10), and 2x2 commuting pairs have an explicit asymptotic with constant 10 zeta(2)/(3 zeta(3)).
Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics
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On commuting integer matrices
Commuting pairs of bounded 3x3 integer matrices are shown to number Theta(N^10), and 2x2 commuting pairs have an explicit asymptotic with constant 10 zeta(2)/(3 zeta(3)).