On the lengths of pairs of complex matrices of size at most five

Abstract

The length of every pair {A, B} of n×n complex matrices is at most 2n − 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2n − 2 span the unital algebra  generated by A, B. For every positive integer m there exist m×m complex matrices C, D such that the length of the pair {C, D} is 2m – 2.