Rationale for Matrix Multiplication in Linear Algebra Textbooks

Abstract

Although matrix multiplication is simple enough to perform, there is reason to believe that it presents conceptual challenges for undergraduate students. For example, it differs from forms of multiplication students with which Linear Algebra students have experience because it is not commutative and does not involve scaling one quantity by another. Rather, matrix multiplication is a multiplication in the sense of abstract algebra: it is associative and distributes over matrix addition. Exposure to abstract algebra’s general treatment of multiplication, however, usually occurs after students have taken Linear Algebra. This elicits the following question: How is matrix multiplication being presented in introductory linear algebra courses? In response, we analyzed the rationale provided for matrix multiplication in 24 introductory Linear Algebra textbooks. We found the ways in which matrix multiplication was explained and justified to be quite varied. In particular, two commonly employed rationalizations are somewhat contradictory, with one approach (isomorphization) suggesting that matrix multiplication can be understood from an early stage, while another (postponement) suggesting that it can only be understood upon consideration of more advanced concepts. We also coordinate these findings with the literature on student thinking in Linear Algebra.