Linear bounds and sharp exponential growth for commuting matrix algebra

For any m pairwise commuting matrices ᵢ\ᵢ₌₁ᵐ⊂ ₙ(), we prove _ [A₁,⋯, Aₘ]≤ (2ᵐ-1)(n-1)+1, answering Bergman's question on the existence of a linear bound for a fixed number m.