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In 1961, Gerstenhaber proved the following theorem: if k is a field and X andY are commuting dxd matrices with entries in k, then the unital k-algebragenerated by these matrices has dimension at most d. The analog of thisstatement for four or more commuting matrices is false. The three matrixversion remains open. We use commutative-algebraic techniques to prove that thethree matrix analog of Gerstenhaber's theorem is true for some new classes ofexamples.  In particular, we translate this three commuting matrix statement into anequivalent statement about certain maps between modules, and prove that thiscommutative-algebraic reformulation is true in special cases. We end with ideasfor an inductive approach intended to handle the three matrix analog ofGerstenhaber's theorem more generally.

New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem