In this paper we propose a novel algorithm called the Bandit-Inspired Memetic Algorithm (BIMA) and we have applied it to solve different large instances of the Quadratic Assignment Problem (QAP). Like other memetic algorithms, BIMA makes use of local search and a population of solutions. The novelty lies in the use of multi-armed bandit algorithms and assignment matrices for generating novel solutions, which will then be brought to a local minimum by local search. We have compared BIMA to multi-start local search (MLS) and iterated local search (ILS) on five QAP instances, and the results show that BIMA significantly outperforms these competitors.