The purpose of the present article is to investigate the definition of matrix multiplication as a central issue in linear algebra courses. Applying both historical and pedagogical approaches, it focuses on the philosophy of generating the usual matrix multiplication, as a special binary operation, with its partly unexpected form compared with the matrix addition as entrywise. Finally, the article examines the reason why the usual matrix multiplication definition has been chosen out of various alternative in the context of school mathematics. It is worth noting that some of the definitions of matrix multiplications have a simpler form to perform, memorize and remember, and are especially more compatible with the individuals' concept images.