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Rationale for Matrix Multiplication in Linear Algebra Textbooks

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Book cover Challenges and Strategies in Teaching Linear Algebra

Part of the book series: ICME-13 Monographs ((ICME13Mo))

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.

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Notes

  1. 1.

    Our descriptions given here are not necessarily identical to those given in each textbook but are instead offered as summaries of these methods that are mathematically equivalent.

  2. 2.

    Methods for multiplying using block/partitioned matrices, if formally addressed in a textbook, typically appeared along with this form of the matrix-matrix product.

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Authors and Affiliations

  1. Oklahoma State University, Stillwater, USA

    John Paul Cook & Adam Estrup

  2. Arizona State University, Tempe, USA

    Dov Zazkis

Authors
  1. John Paul Cook
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  2. Dov Zazkis
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  3. Adam Estrup
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Correspondence to John Paul Cook .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA

    Sepideh Stewart

  2. College of Education, Florida State University, Tallahassee, Florida, USA

    Christine Andrews-Larson

  3. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

    Avi Berman

  4. Arizona State University, Mesa, Arizona, USA

    Michelle Zandieh

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Cook, J.P., Zazkis, D., Estrup, A. (2018). Rationale for Matrix Multiplication in Linear Algebra Textbooks. In: Stewart, S., Andrews-Larson, C., Berman, A., Zandieh, M. (eds) Challenges and Strategies in Teaching Linear Algebra. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-66811-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-66811-6_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66810-9

  • Online ISBN: 978-3-319-66811-6

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