Maths lecturers in denial about their own maths practice? A case of teaching matrix operations to undergraduate students


  • Alexander Partner University of Essex
  • Alexei Vernitski University of Essex



matrices, higher education, research and teaching practice, educational material and media


This case study provides evidence of an apparent disparity in the way that certain mathematics topics are taught compared to the way that they are used in professional practice. In particular, we focus on the topic of matrices by comparing sources from published research articles against typical undergraduate textbooks and lecture notes. Our results show that the most important operation when using matrices in research is that of matrix multiplication, with 33 of the 40 publications which we surveyed utilising this as the most prominent operation and the remainder of the publications instead opting not to use matrix multiplication at all rather than offering weighting to alternative operations. This is in contrast to the way in which matrices are taught, with very few of these teaching sources highlighting that matrix multiplication is the most important operation for mathematicians. We discuss the implications of this discrepancy and offer an insight as to why it can be beneficial to consider the professional uses of such topics when teaching mathematics to undergraduate students.

Author Biographies

Alexander Partner, University of Essex

Research Student, Mathematics Education

Alexei Vernitski, University of Essex



Amanatidis, G., Green, B. and Mihail, M., 2018. Connected realizations of joint-degree matrices. Discrete Applied Mathematics, 250, pp.65–74. doi:

Amanatidis, G. and Kleer, P., 2018. Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, pp.966–985. Available at:

Arrigo, F. Grindrod, P., Higham, D.J. and Noferini, V., 2018a. Non-backtracking walk centrality for directed networks. Journal of Complex Networks, 6(1), pp.54–78. doi:

Arrigo, F. Grindrod, P., Higham, D.J. and Noferini, V., 2018b. On the exponential generating function for non-backtracking walks. Linear Algebra and its Applications, 556, pp.381–399. doi:

Aslanyan, V., Eterović, S. and Kirby, J., 2021. Differential Existential Closedness for the j-function. In Proceedings of the American Mathematical Society, 149(4), pp.1417–1429. doi:

Basios, V., Antonopoulos, C.G. and Latifi, A., 2020. Labyrinth chaos: Revisiting the elegant, chaotic and hyperchaotic walks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(11), p. 113129. doi:

Blackburn, S.R. and Claridge, J., 2019. Finite-Field Matrix Channels for Network Coding. IEEE Transactions on Information Theory, 65(3), pp.1614–1625. doi:

Brandt, M., Dipper, R., James, G. and Lyle, S., 2009. Rank polynomials. Proceedings of the London Mathematical Society, 98(1), pp.1–18. doi:

Chopra, K., Hodges, H.R., Barker, Z.E., Vázquez Diosdado, J.A., Amory, J.R., Cameron, T.C., Croft, D.P., Bell, N.J. and Codling, E.A., 2020. Proximity Interactions in a Permanently Housed Dairy Herd: Network Structure, Consistency, and Individual Differences. Frontiers in Veterinary Science, 7, p. 583715. doi:

Claridge, J. and Chatzigeorgiou, I., 2017. Probability of Partially Decoding Network-Coded Messages. IEEE Communications Letters, 21(9), pp.1945–1948. doi:

De Boeck, M., Evseev, A., Lyle, S. and Speyer, L., 2018. On Bases of Some Simple Modules of Symmetric Groups and Hecke Algebras. Transformation Groups, 23(3), pp.631–669. doi:

Ding, L., Yu, D., Xie, J., Guo, W., Hu, S., Liu, M., Kong, L., Dai, H., Bao, Y. and Jiang, B., 2021. Word Embeddings via Causal Inference: Gender Bias Reducing and Semantic Information Preserving. Available at:

Dolinka, I. and Gray, R., 2013. Maximal subgroups of free idempotent generated semigroups over the full linear monoid. Transactions of the American Mathematical Society, 366(1), pp.419–455. doi:

Dolinka, I. and Gray, R.D., 2018. Universal locally finite maximally homogeneous semigroups and inverse semigroups. Forum Mathematicum, 30(4), pp.947–971. doi:

Dolinka, I., Gray, R.D. and Ruškuc, N., 2017. On regularity and the word problem for free idempotent generated semigroups: The World Problem for Free Idempotent Generated Semigroups. In Proceedings of the London Mathematical Society, 114(3), pp.401–432. doi:

Fayers, M. and Lyle, S., 2009. Some reducible Specht modules for Iwahori–Hecke algebras of type A with q=−1. Journal of Algebra, 321(3), pp.912–933. doi:

Fayers, M. and Lyle, S., 2013. The reducible Specht modules for the Hecke algebra H_(C-1) (S_n ). Journal of Algebraic Combinatorics, 37(2), pp.201–241. doi:

Gray, R.D., 2014. The minimal number of generators of a finite semigroup. Semigroup Forum, 89(1), pp.135–154. doi:

Gray, R.D. and Kambites, M., 2020. On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup. International Mathematics Research Notices, 2020(12), pp.3753–3793. doi:

Grindrod, P., Higham, D.J. and Noferini, V., 2018. The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis. SIAM Journal on Matrix Analysis and Applications, 39(1), pp.310–341. doi:

Hadjiantoni, S., 2022. An alternative numerical method for estimating large-scale time-varying parameter seemingly unrelated regressions models. Econometrics and Statistics, 21, pp.1–18. doi:

Hadjiantoni, S. and Kontoghiorghes, E.J., 2018. A recursive three-stage least squares method for large-scale systems of simultaneous equations. Linear Algebra and its Applications, 536, pp.210–227. doi:

Kirby, J., 2010. Exponential algebraicity in exponential fields. Bulletin of the London Mathematical Society, 42(5), pp.879–890. doi:

Kirby, J., 2016. The rational field is not universally definable in pseudo-exponentiation’, Fundamenta Mathematicae, 232(1), pp.79–88. doi:

Lameu, E.L., Borges, F.S., Iarosz, K.C., Protachevicz, P.R., Antonopoulos, C.G., Macau, E.E.N. and Batista, A.M., 2021. Short-term and spike-timing-dependent plasticity facilitate the formation of modular neural networks. Commun Nonlinear Sci Numer Simulat, 96, p. 105689. doi:

Liu, F. and Siemons, J., 2022. Unlocking the walk matrix of a graph. Journal of Algebraic Combinatorics, 55(3), pp.663–690. doi:

Liu, F., Siemons, J. and Wang, W., 2019. New families of graphs determined by their generalized spectrum. Discrete Mathematics, 342(4), pp.1108–1112. doi:

Lyle, S., 2007. Some q-analogues of the Carter-Payne theorem. Journal fur die reine und angewandte Mathematik, 608, pp.93–121. doi:

Lyle, S., 2013. On Homomorphisms Indexed by Semistandard Tableaux. Algebra Represent Theory, 16, pp.1409–1447. doi:

Mazorchuk, V. and Miemietz, V., 2011. Additive versus abelian 2-representations of fiat 2-categories. Available at:

Mazorchuk, V. and Miemietz, V., 2015. Transitive 2-representations of finitary 2-categories. Transactions of the American Mathematical Society, 368(11), pp.7623–7644. doi:

Mazorchuk, V. and Miemietz, V., 2016. Isotypic faithful 2-representations of J-simple fiat 2-categories. Mathematische Zeitschrift, 282(1–2), pp.411–434. doi:

Mazorchuk, V., Miemietz, V. and Zhang, X., 2019. Pyramids and 2-representations. Revista Matemática Iberoamericana, 36(2), pp.387–405. doi:

Mehrmann, V., Noferini, V., Tisseur, F. and Xu, H., 2016. On the sign characteristics of Hermitian matrix polynomials. Linear Algebra and its Applications, 511, pp.328–364. doi:

Noferini, V., 2012. The behaviour of the complete eigenstructure of a polynomial matrix under a generic rational transformation. The electronic journal of linear algebra ELA, 23(1). doi:

Noferini, V., Sharify, M. and Tisseur, F., 2015. Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 36(1), pp.138–157. doi:

Noferini, V. and Williams, G., 2021. Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds. Journal of Algebra, 587, pp.1–19. doi:

Smith, Q.M., Inchingolo, A.V., Mihailescu, M., Dai, H. and Kad, N.M., 2021. Single-¬molecule imaging reveals the concerted release of myosin from regulated thin filaments. eLife, 10, p. e69184. doi:

Vernitski, A., 2007. A Generalization of Symmetric Inverse Semigroups. Semigroup Forum, 75(2), pp.417–426. doi:

Williams, G., 2014. Smith forms for adjacency matrices of circulant graphs. Linear Algebra and its Applications, 443, pp.21–33. doi:

The Open University, 2006. 208 Pure Mathematics - Linear equations and matrices: LA2. Available at: (Accessed: 30 May 2020).

Aguilar, C.O., n.d. MATH 233 - Linear Algebra I. Department of Mathematics, SUNY Geneso New York.

Al-Azemi, A., 2017. Lecture Notes in Linear Algebra. Mathematics Department - Kuwait University.

Axler, S., 2015. Linear Algebra Done Right. Cham: Springer International Publishing (Undergraduate Texts in Mathematics). Available at: (Accessed: 30 May 2022).

Beezer, R.A., 2015. A first course in linear algebra. Gig Harbor, Wash.: Congruent Press. Available at: (Accessed: 30 May 2022).

Boyd, S.P. and Vandenberghe, L., 2018. Introduction to applied linear algebra: vectors, matrices, and least squares. Cambridge, UK ; New York, NY: Cambridge University Press.

Bright, M. and Krammer, D., 2011. MA106 Linear Algebra lecture notes. University of Warwick.

Bronson, R. and Costa, G.B., 2007. Linear algebra: an introduction. 2nd ed. Amsterdam ; Boston: Elsevier.

Cameron, P.J., 2008. Linear Algebra. Queen Mary University London.

Carey, 1998. Introduction to Matrix Algebra. Psychology 7291 University of Colorado.

Carrell, J.B., 2005. Fundamentals of Linear Algebra. The University of British Columbia.

Chandra, P., Lal, A.K., Raghavendra, V. and Santhanam, G., n.d. Notes on Mathematics 102. Indian Institute of Technology Kanpur.

Cook, J.S., 2015. Lecture Notes for Linear Algebra. Department of Mathematics - Liberty University.

Cooperstein, B., 2016. Elementary Linear Algebra. University of California, Santa Cruz.

Cresswell, M., 2006. AQA Further Pure 4. Manchester: AQA.

Dawkins, P., 2005. Linear Algebra. Cornell University.

Denton, T. and Waldron, A., 2012a. Linear Algebra in Twenty Five Lectures.

Dunn, F. and Parberry, I., 2002. 3D Math Primer for Graphics and Game Development. Jones & Bartlett Publishers.

Earl, R., 2021. Linear Algebra I. University of Oxford.

Simon Fraser University, n.d. ECON 331 Lecture Notes. Department of Economics.

Gunawardena, J., 2006. Matrix algebra for beginners, Part I. Department of Systems Biology, Havard Medical School.

Hartman, G.N., 2011. Fundamentals of matrix algebra. APEX Calculus.

Hefferon, J., 2008. Linear Algebra.

Kunze, R., 1971. Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall.

Kuttler, K., 2012. Linear Algebra: Theory and Applications. The Saylor Foundation.

Kuttler, K. and Farah, I., 2017. A first course in linear algebra. Lyrynx.

Lang, S., 2012. Introduction to linear algebra. Springer Science & Business Media.

Langley, P.J.K., n.d. Applied Algebra MTHS2002. University of Nottingham.

Larson, R. and Falvo, D.C., 2009. Elementary linear algebra. 6th ed. Boston: Houghton Mifflin Harcourt Pub. Co.

University of Arizona, 2012. Lecture 1: Intro or Refresher in Matrix.

Utrecht University, 2012. Lecture 4: matrices, determinants.

Lerner, D., 2008. Lecture notes on linear algebra. University of Kansas.

Lipschutz, S. and Lipson, M., 2011. Schaum’s outlines: linear algebra. New York: McGraw Hill Professional.

Margalit, D., Rabinoff, J. and Rolen, L., 2017. Interactive Linear Algebra. Georgia Institute of Technology.

Noferini, V., 2017. MA114: Linear Mathematics. University of Essex.

von Schlippe, W.B., n.d. Mathematical Techniques Part 4: Matrix Algebra. Department of Psychology, Saint Petersburg University.

Selinger, P., n.d. Matrix theory and linear algebra. Lyrynx.

Smith, H., 2017. Core Pure Mathematics Book 1/AS. Pearson Education Ltd.

Tao, T., 2002. Lecture notes for Math 115a (Linear Algebra). UCLA.

Birkhoff, G. and MacLane, S., 2017. A survey of modern algebra. AK Peters/CRC Press.

Bourbaki, N.,1958. Algèbre. Livre 2, ch. 2, Hermann.

Chollet, F., 2017. Deep learning with Python. Manning Publications Co.

Harari, Y.N., 2018. 21 Lessons for the 21st Century. Random House.

Olver, P.J., and Shakiban, C., 2006. Applied linear algebra (Vol. 1). Upper Saddle River, NJ: Prentice-Hall.

Partner, A., 2020. Matrix Multiplication (1 of 3: Contextualising the Process) (Accessed: 13 June 2022)

Woo, E., 2014. Matrix Multiplication (1 of 3: Basic Principles) (Accessed: 30 May 2022)


Additional Files