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We introduce an equivalence relation  ρ  on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving  ρ . Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves  ρ . Finally, we illustrate that there exist non-surjective linear transformations that preserve  ρ  or preserve Kleene stars of matrices or commute with the Kleene star operator.

Linear preservers for matrices over a class of semirings