Abstract: What is the dimension of a triply generated commutative matrix algebra? It seems that not much is known, but we’ll discuss some relevant ideas. For example, an old result, often called Gerstenhaber’s Theorem, states that the algebra of polynomials in two commuting nxn matrices has dimension at most n. Here we discuss the possibility of extending this result to algebras generated by three commuting matrices. Related questions concern the reducibility of the variety of commuting triples and the question of “approximate simultaneous diagonalizability”. We present some experimental results based on the Weyr canonical form (an under–appreciated alternative to the JCF).