NOTE ON THE HISTORY OF (SQUARE) MATRIX AND DETERMINANT

  • Olayiwola Babarinsa Federal University Lokoja https://orcid.org/0000-0002-3569-0828
  • Azfi Zaidi Mohammad Sofi , Universiti Malaysia Kelantan
  • Asrul Hery Mohd Universiti Malaysia Kelantan
  • Akinola Eluwole Federal University Oye-Ekiti, P.M.B 373, Oye, Ekiti, Nigeria
  • Imoni Sunday Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Wakili Adamu Federal University Lokoja
  • Benson Onojhojobi Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Momoh Sheidu Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Rotimi Kehinde Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Lanlege Daniel Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Shobanke Dolapo Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Helen Olaronke Edogbanya Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Anselm Oyem Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Isaac Adeniyi Federal University Lokoja, P.M.B 1154, Kogi
  • Chinenye Ezenweke Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Sani Umaru Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Sabastine Emmanuel
  • Kefas Bitrus
  • Veronica Cyril-Okeme,
  • Eunice Upahi
  • Emmanuel Akaligwo Akaligwo Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Luke Arinze
  • Simon Barguma Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Friday Edibo Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Mayowa Atteh Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Alloy Idoko Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Enoch Opeyemi Federal University Oye-Ekiti, P.M.B 373, Oye, Ekiti, Nigeria
  • Mogbademu Adesanmi University of Lagos, Akoka, Lagos, Nigeria
  • Olaitan Ojo University of New Haven, 300 Boston Post Road, Connecticut, U.S.A
  • Akeem Disu National Open University of Nigeria, Abuja, Nigeria
  • Tajudeen Adeeko University of Abuja, P.M.B 117, Abuja, Nigeria
  • Geraldine Anukwu University of Lagos, Akoka, Lagos, Nigeria
  • Damilola Samson
  • Kunle Ogunleye Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • Jude Koffa Federal University Lokoja, P.M.B 1154, Kogi, Nigeria
  • ojonubah James Federal College of Education Okene, P.M.B 1026, Kogi, Nigeria
  • Marut Musa University of Jos
  • Niri Martha Choji Plateau State University, Bokkos, Jos, Nigeria
  • Helen Oluyemisi Emeka Afe Babalola University, Ado, Ekiti, Nigeria
  • Adamu Umar Faculty of Physical Sciences, Ahmadu Bello University, P.M.B 1045, Zaira-Kaduna, Nigeria
  • Mansur Hassan Yusuf Maitama Sule University
  • Oluwaseyi Jaiyeoba Purdue University, West Lafayette, Indiana USA
  • Shamsoudine Aidara Universiti Malaysia Kelantan, 16100 Kota Bharu, Malaysia
  • Mamuda Mamman Niger State College of Education, Minna, Niger, Nigeria
  • Edna Manga University of Jos, P.M.B 2084, Jos, Nigeria
  • Amiru Sule Federal University Gusau, Gusau, Zamfara, Nigeria
  • Osagie Uyimwen University of Abuja, P.M.B 117, Abuja, Nigeria
  • Abdullah-al-Musa Ahmed Bangladesh University of Business and Technology, Rupnagar Mirpur-2, Dhaka-1216, Bangladesh
  • Zaku Garba Plateau State University, Bokkos, Jos, Nigeria
  • Zaato Gbene Universiti Malaysia Kelantan, 16100 Kota Bharu, Malaysia
  • Sanusi Jubri Kano University of Science and Technology, Wudil, Kano, Nigeria
  • Godwin Okeke School of Physical Sciences, Federal University of Technology, Owerri, Imo, Nigeria
Keywords: Matrix, determinant, linear systems, history of mathematics

Abstract

This paper reviews the theory of matrices and determinants. Matrix and determinant are nowadays considered inseparable to some extent, but the determinant was discovered over two centuries before the term matrix was coined. Our review associate determinant with the matrix as part of linear systems but not with polynomials. Thus, the paper first gives the background on matrix with vast applications in all fields of study and then reviews the history of determinants which is based on its major contributors in chronological order from the sixteenth century to the twenty-first century

Author Biographies

Olayiwola Babarinsa, Federal University Lokoja

Department of Mathematics

Azfi Zaidi Mohammad Sofi, , Universiti Malaysia Kelantan

Faculty of Bioengineering and Technology,

Asrul Hery Mohd, Universiti Malaysia Kelantan

Faculty of Bioengineering and Technology,

Akinola Eluwole, Federal University Oye-Ekiti, P.M.B 373, Oye, Ekiti, Nigeria

Deparment of Geophysics

Imoni Sunday, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematics

Wakili Adamu , Federal University Lokoja

Department of Mathematics

Benson Onojhojobi, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Statistics

Momoh Sheidu, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematics

Rotimi Kehinde, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematics

Lanlege Daniel, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematic

Shobanke Dolapo, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Statistics

Helen Olaronke Edogbanya, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematics

Anselm Oyem, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Mathematics

Isaac Adeniyi, Federal University Lokoja, P.M.B 1154, Kogi

Department of Statistics

Chinenye Ezenweke, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Statistics

Simon Barguma, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Statistics

Enoch Opeyemi, Federal University Oye-Ekiti, P.M.B 373, Oye, Ekiti, Nigeria

Department of Mathematics

Mogbademu Adesanmi , University of Lagos, Akoka, Lagos, Nigeria

Department of Mathematics

Olaitan Ojo, University of New Haven, 300 Boston Post Road, Connecticut, U.S.A

Pompea College of Business

Akeem Disu, National Open University of Nigeria, Abuja, Nigeria

Department of Mathematics,

Tajudeen Adeeko , University of Abuja, P.M.B 117, Abuja, Nigeria

Department of Physics,

Geraldine Anukwu, University of Lagos, Akoka, Lagos, Nigeria

Department of Geosciences

Damilola Samson

Department of Physics,

Kunle Ogunleye, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Physics,

Jude Koffa, Federal University Lokoja, P.M.B 1154, Kogi, Nigeria

Department of Physiscs

ojonubah James, Federal College of Education Okene, P.M.B 1026, Kogi, Nigeria

Department of Mathematics,

Marut Musa, University of Jos

Department of Mathematics

Niri Martha Choji, Plateau State University, Bokkos, Jos, Nigeria

Department of Mathematics

Helen Oluyemisi Emeka, Afe Babalola University, Ado, Ekiti, Nigeria

Department of Mathematical and Physical Sciences

Adamu Umar, Faculty of Physical Sciences, Ahmadu Bello University, P.M.B 1045, Zaira-Kaduna, Nigeria

Department of Statistics

Mansur Hassan, Yusuf Maitama Sule University

Department of Mathematics

Oluwaseyi Jaiyeoba, Purdue University, West Lafayette, Indiana USA

Department of Computer Graphics Technology,

Shamsoudine Aidara, Universiti Malaysia Kelantan, 16100 Kota Bharu, Malaysia

Faculty of Entrepreneurship and Business

Mamuda Mamman, Niger State College of Education, Minna, Niger, Nigeria

Department of Mathematics

Edna Manga, University of Jos, P.M.B 2084, Jos, Nigeria

Department of Mathematics

Amiru Sule, Federal University Gusau, Gusau, Zamfara, Nigeria

Department of Mathematics

Osagie Uyimwen, University of Abuja, P.M.B 117, Abuja, Nigeria

Department of Physics

Abdullah-al-Musa Ahmed, Bangladesh University of Business and Technology, Rupnagar Mirpur-2, Dhaka-1216, Bangladesh

Computer Sciences and Engineering Department

Zaku Garba, Plateau State University, Bokkos, Jos, Nigeria

Department of Mathematics

Zaato Gbene, Universiti Malaysia Kelantan, 16100 Kota Bharu, Malaysia

Faculty of Entrepreneurship and Business

Sanusi Jubri, Kano University of Science and Technology, Wudil, Kano, Nigeria

Department of Statistics

Godwin Okeke, School of Physical Sciences, Federal University of Technology, Owerri, Imo, Nigeria

Department of Mathematics

References

Abeles, F. (1986). Determinants and linear systems: Charles L. Dodgson's view. The British Journal for the history of science, 19(03), 331-335.
Abeles, F. (1994). The mathematical pamphlets of Charles Lutwidge Dodgson and related pieces. Lewis Carroll Society of North America.
Abeles, F. (2008). Dodgson condensation: The historical and mathematical development of an experimental method. Linear Algebra and its Applications, 429(2-3), 429-438. doi:10.1016/j.laa.2007.11.022
Abeles, F. (2014). Chiò's and Dodgson's determinantal identities. Linear Algebra and its Applications, 454, 130-137. doi:http://dx.doi.org/10.1016/j.laa.2014.04.010
Afriat, S. (2000). Determinants and Matrices. Linear Dependence, Springer, Boston.
Ahmed, A. A. and Bondar, K. (2014). Modern method to compute the determinants of matrices of order 3. Journal of Informatics and Mathematical Sciences, 6(2), 55-60.
Aitken, A. C. (1956). Determinants and matrices. Edinburgh; New York: Oliver and Boyd; Interscience Publishers.
Akritas, A. G., Akritas, E. K. and Malaschonok, G. I. (1996). Various proofs of Sylvester's (determinant) identity. Mathematics and Computers in Simulation, 42(4–6), 585-593. doi:http://dx.doi.org/10.1016/S0378-4754(96)00035-3
Almalki, S., Alzahrani, S. and Alabdullatif, A. (2013). New parallel algorithms for finding determinants of N× N matrices. Paper presented at the Computer and Information Technology (WCCIT), 2013 World Congress on.
Amdeberhan, T. (2001). Determinants through the Looking Glass. Advances in Applied Mathematics, 230, 225-230. doi:10.1006/aama.2001.0732
Amdeberhan, T. and Ekhad, S. B. (1997). A Condensed Condensation Proof of a Determinant Evaluation Conjectured by Greg Kuperberg and Jim Propp. Journal of Combinatorial Theory. Series A 78, 169-170.
Athloen, S. and McLaughlin, R. (1987). Gauss-Jordan reduction: A brief history. American
Mathematical Monthly 94, 130-142.
Babarinsa, O. (2020). Algebra in African Indigenous History. In A. Nhemachena, N. Hlabangane, and J. Matowanyika (Eds.), Decolonising Science, Technology, Engineering and Mathematics (STEM) in an Age of Technocolonialism: Recentring African Indigenous Knowledge and Belief Systems (pp. 199-212). Cameroon: Langaa RPCIG.
Babarinsa, O., Arif, M. and Kamarulhaili, H. (2019). Potential applications of hourglass matrix and its quadrant interlocking factorization. ASM Science Journal, 12(5), 72 - 79.
Bareiss, E. H. (1968). Sylvester’s identity and multistep integer-preserving Gaussian elimination. Mathematics of computation, 22(103), 565-578.
Bell, E. T. (2014). Men of mathematics: Simon and Schuster.
Benzi, M. (2009). The Early History of Matrix Iterations: With a Focus on the Italian Contribution. Paper presented at the SIAM Conference on Applied Linear Algebra, Monterey Bay, Seaside, California.
Benzi, M. (2009). Key moments in the history of numerical analysis. Paper presented at the SIAM Applied Linear Algebra Conference.
Berkhout, A. (2008). Signal Models in Seismic Processing. Handbook of Signal Processing in Acoustics (pp. 1559-1570): Springer.
Bernardes, A. and Roque, T. (2018). History of matrices Mathematics, Education and History: Springer. pp. 209-227
Bernstein, D. S. (2009). Matrix mathematics: theory, facts, and formulas. Princeton University Press.
Bézout, E. (1779). Théorie générale des équations algébriques; par m. Bézout: de l'imprimerie de Ph.-D. Pierres, rue S. Jacques.
Bézout, É. (1762). Sur plusieurs classes d’équations de tous les degrés qui admettent une solution algébrique. Histoire de l’Académie royale des sciences, partie Mémoires, 17-52.
Bôcher, M. and Duval, E. P. R. (1922). Introduction to higher algebra. Macmillan.
Bonolis, L. and de Laplace, P. S. (2004). From the Rise of the Group Concept to the Stormy Onset of Group Theory in the New Quantum Mechanics. A saga of the invariant characterization of physical objects, events, and theories. Rivista del Nuovo Cimento, 27(4-5), 39.
Boyer, C. B. (1966). Colin Maclaurin and Cramer’s rule. Scripta Mathematica, 27(4), 377-379.
Bressoud, D. and Propp, J. (1999). How the alternating sign matrix conjecture was solved. Notices-American Mathematical Society, 46(i), 637-646.
Bronson, R. (1988). Schaum's outline of theory and problems of matrix operations. New York: McGraw-Hill.
Browne, E. T. (2018). Introduction to the Theory of Determinants and Matrices. UNC Press Books.
Brualdi, R. A. and Schneider, H. (1983). Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley. Linear Algebra and its Applications, 52-53(C), 769-791.
Brunetti, M. and Renato, A. (2014). Old and New Proofs of Cramer’ s Rule. Applied Mathematical Sciences, 8(133), 6689-6697.
Burton, D. M. (2003). The history of mathematics. Mc Graw Hill.
Bylina, B. and Bylina, J. (2009). Influence of preconditioning and blocking on accuracy in solving Markovian models. International Journal of Applied Mathematics and Computer Science, 19(2), 207-217.
Campbell, H. G. (1980). Linear algebra with applications: Prentice Hall.
Cardano, G. (1993). Ars Magna or The Rules of Algebra: Transl. and ed. Retrieved from Dover: New York.
Cardano, G. and Spon, C. (1968). Ars magna (1545). Opera Omnia, 4, 221-302.
Cardano, G., Witmer, T. R. and Ore, O. (2007). The Rules of Algebra: Ars Magna. Courier Corporation., New York.
Cauchy, A. L. (1812). Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment. Journal de l’Ecole polytechnique, 17, 29-112.
Cayley, A. (1845). On the theory of linear transformations. Cambridge Math. J, 4(1845), 1-16.
Cayley, A. (1858). A memoir on the theory of matrices. Philosophical transactions of the Royal society of London, 148, 17-37.
Chang, F. C. (2014). Determinant of matrix by order condensation. Theory and Applications of Mathematical Science,1, 73-78.
Chang, F. C. and Su, C. T. (1998). More on quick evaluation of determinants. Applied Mathematics and Computation, 93(1), 97-99. doi:http://dx.doi.org/10.1016/S0096-3003(97)10044-3
Channell, R. E. (1977). A compendium, the women of mathematics. (Master of Arts), Emporia State University, Kansas.
Chapra, S. C. and Canale, R. P. (1998). Numerical methods for engineers (Vol. 2). McGraw-Hill.
Chen, X. B. (2014). A fast algorithm for computing the determinants of banded circulant matrices. Applied Mathematics and Computation, 229, 201-207. doi:10.1016/j.amc.2013.12.048
Churchhouse, R. F. (2002). Codes and ciphers: Julius Caesar, the Enigma, and the Internet. Cambridge University Press.
Cinkir, Z. (2014). A fast elementary algorithm for computing the determinant of Toeplitz matrices. Journal of Computational and Applied Mathematics, 255, 353-361. doi:10.1016/j.cam.2013.05.014
Cormen, T. H. (2009). Introduction to algorithms: Cambridge, Mass. MIT Press.
Cramer, G. (1750). Introduction à l'analyse des lignes courbes algébriques. Europeana, 656-659.
D’Andrea, A., Ferri, F. and Grifoni, P. (2010). An overview of methods for virtual social networks analysis. Computational social network analysis, 3-25.
Debnath, L. (2013a). A brief historical introduction to determinant with applications. International Journal of Mathematical Education in Science and Technology, 44(3), 388-407.
Debnath, L. (2013b). The modern origin of matrices and their applications. International Journal of Mathematical Education in Science and Technology, 45(4), 528-551. doi:10.1080/0020739x.2013.851808
Degos, J. G. (2015). Brief History of Matrices, As a Tool of Consolidated Financial Statements. Muhasebe ve Finans Tarihi Araştırmaları Dergisi, 8: 51-78.
Descartes, R. (1886). La géométrie de René Descartes, vol. 1. Hermann.
Djungu, S. J. A. and Manneback, P. (2020). SpeedSiteRank: PageRank algorithm distributed in websites. International Journal of Computer Science Issues, 17(2): 13-18.
Dossey, J. A., Otto, A. D., Spence, L. E. and Eynden, C. V. (2001). Discrete mathematics. Harlow: Addison Wesley.
Dunham, C. B. (1980). Cramer's rule reconsidered or equilibration desirable. ACM SIGNUM Newsletter, 15(4), 9-9.
Eberly, D. H. (2001). 3D game engine design : a practical approach to real-time computer graphics. San Francisco: Morgan Kaufmann.
El-Mikkawy, M. E. (2008). A fast and reliable algorithm for evaluating nth order pentadiagonal determinants. Applied Mathematics and Computation, 202(1), 210-215.
Eves, H. (1969). An introduction to the history of mathematics. New York: Holt, Rinehart and Winston.
Eves, H. W. (1980). Elementary matrix theory: Courier Corporation.
Feng, Q. and Zhou, Y. (2014). Soft discernibility matrix and its applications in decision making. Applied Soft Computing, 24, 749-756.
Francisco Neto, A. (2015). A note on a determinant identity. Applied Mathematics and Computation, 264, 246-248. doi:http://dx.doi.org/10.1016/j.amc.2015.04.079
Franklin, J. N. (1968). Matrix theory. Englewood Cliffs, N.J.: Prentice-Hall.
Gauss, C. F. (1966). Disquisitiones arithmeticae (Vol. 157). Yale University Press.
Godin, L., Demours, P. and Cotte, L. (1774). Table alphabétique des matières contenues dans l'Histoire et les Mémoires de l'Académie Royale des Sciences (Vol. 8): Comp. des libraires.
Grcar, J. F. (2011). How ordinary elimination became Gaussian elimination. Historia Mathematica, 38(2), 163-218.
Grcar, J. F. (2012). Review of The Chinese Roots of Linear Algebra by Roger Hart. Bull. Amer. Math. Soc, 49(4), 589.
Gu, C. and Xu, Z. (2008). Condensed cramer rule for computing a kind of restricted matrix equation. Journal of Applied Mathematics and Informatics, 26(5), 1011-1020.
Günther, S. (1908). Geschichte der Mathemat. G.J. Göschen. Berlin.
Gutman, I. (1977). Acyclic systems with extremal Hückel π-electron energy. Theoretica chimica acta, 45(2), 79-87.
Habgood, K. and Arel, I. (2010). Revisiting Cramer's rule for solving dense linear systems. In the Proceedings of the 2010 Spring Simulation Multiconference.
Habgood, K. and Arel, I. (2012). A condensation-based application of Cramerʼs rule for solving large-scale linear systems. Journal of Discrete Algorithms, 10, 98-109. doi:10.1016/j.jda.2011.06.007
Hadamard, J. (1897). Mémoire sur l'élimination. Acta mathematica, 20(1), 201-238.
Hart, R. (2011). The Chinese roots of linear algebra. Baltimore: The John Hopkins University Press.
Hawe, P., Webster, C. and Shiell, A. (2004). A glossary of terms for navigating the field of social network analysis. Journal of epidemiology and community health, 58(12), 971-975.
Hawkins, T. (1974). The theory of matrices in the 19th century. In the Proceedings of the international congress of mathematicians, Vancouver.
Hedman, B. A. (1999). An Earlier Date for “Cramer's Rule”. Historia Mathematica, 26(4), 365-368.
Henry, N. and Fekete, J. D. (2007). Henry, Nathalie, and Jean-Daniel Fekete. Matlink: Enhanced matrix visualization for analyzing social networks. In IFIP Conference on Human-Computer Interaction, pp. 288-302.
Higham, N. J. (2002). Accuracy and stability of numerical algorithms. Siam.
Hoffman, J. D. and Frankel, S. (2001). Numerical methods for engineers and scientists. CRC press.
Horn, R. A. and Johnson, C. R. (2012). Matrix analysis. Cambridge university press.
Ivanov, S., Ivanova, L. and Meleshkova, Z. (2020). Calculation and Optimization of Industrial Robots Motion. In the Proceedings of the 2020 26th Conference of Open Innovations Association (FRUCT).
Jacobi, C. G. J. (1896). Ueber die bildung und die eigenschaften der determinanten:(De formatione et proprietatibus determinantium.): W. Engelmann.
Jaffe, A. (1984). Ordering the universe: the role of mathematics. SIAM Review, 26(4), 473-500.
Janjia, M. (2005). A note on Laplace's expansion theorem. International Journal of Mathematical Education in Science and Technology, 36(6), 696-698.
Jeffrey, A. (2010). Matrix Operations for Engineers and Scientists: An Essential Guide in Linear Algebra. Springer Science and Business Media.
Ji, J. (2012). A condensed Cramer’s rule for the minimum-norm least-squares solution of linear equations. Linear Algebra and its Applications, 437(9), 2173-2178.
Kani, E. (2011). Idoneal numbers and some generalizations. Mathematics Annales mathématiques du Québec, 35(2), 197-227.
Karim, S. (2013). New Sequential and Parallel Division Free Methods for Determinant of matrices, Ph.D thesis, Universiti Utara Malaysia, Malaysia.
Karim, S., Ibrahim, H. and Omar, Z. (2016). Some modifications of Sarrus’s rule method via permutation for finding determinant of 4 by 4 square matrix. In the Proceedings of the AIP Conference 2016.
Kariuki, S. and Löwe, K. (2007). Integrating human factors into process hazard analysis. Reliability Engineering and System Safety, 92(12), 1764-1773.
Kendall, D. G. (1949). Stochastic processes and population growth. Journal of the Royal Statistical Society. Series B (Methodological), 11(2), 230-282.
Kippenhahn, R. (1999). Code breaking : History and exploration. Universities Press.
Kline, M. (1990). Mathematical thought from ancient to modern times (Vol. 3). Oxford University Press.
Klinger, A. (1967). The Vandermonde Matrix. The American Mathematical Monthly, 74(5), 571-574. doi:10.2307/2314898
Knobloch, E. (1994). From Gauss to Weierstrass: determinant theory and its historical evaluations, The intersection of history and mathematics (pp. 51-66): Springer.
Knobloch, E. (2013). Leibniz’s Theory of Elimination and Determinants. In Seki, Founder of Modern Mathematics in Japan. Springer, Tokyo, pp. 229-244
Kosinski, A. (2001). Cramer's rule is due to Cramer. Mathematics Magazine, 74(4), 310-312.
Kronecker, L. (1903). Vorlesungen über die Theorie der Determinanten, Erster Band, Bearbeitet und fortgeführt von K. Hensch, BG Teubner, Leipzig.
Kruh, L. and Deavours, C. (2002). The commercial enigma: beginnings of machine cryptography. Cryptologia, 26(1), 1-16.
Kyrchei, I. (2008). Cramer's rule for quaternionic systems of linear equations. Journal of Mathematical Sciences, 155(6), 839-858.
Kyrchei, I. I. (2015). Cramer's rule for generalized inverse solutions. Advances in Linear Algebra Research, Nova Sci. Publ., New York, 79-132.
Lagrange, J. L. (1775). Recherches d'arithmetique: Nouveaux Mémoires de l’Académie de Berlin,
Lancaster, P. and Tismenetsky, M. (1985). The theory of matrices: With applications. Orlando: Academic Press.
Lefkovitch, L. (1965). The study of population growth in organisms grouped by stages. Biometrics, 1-18.
Leggett, D., Perry, J. E. and Torrence, E. (2009). Generalizing Dodgson's method: a" double-crossing" approach to computing determinants. arXiv preprint arXiv: …, 1-14. doi:10.4169/college.math.j.42.1.043
Lengyel, E. (2012). Mathematics for 3D game programming and computer graphics. Boston: Course Technology PTR.
Lenhart, S. M. and Travis, C. C. (1986). Global stability of a biological model with time delay. In the Proceedings of the American Mathematical Society, pp: 75-78.
MacBeth, C. and Li, X. Y. (1996). Linear matrix operations for multicomponent seismic processing. Geophysical Journal International, 124(1), 189-208.
MacDuffee, C. (1934). The theory of matrices. Bull. Amer. Math. Soc, 40, 372-373.
MacLaurin, C. (1748). A treatise of algebra. London: A. Millar, and J. Nourse.
Mangal, A., Malik, H. and Aggarwal, G. (2020). An Efficient Convolutional Neural Network Approach for Facial Recognition. In the proceedings of the 2020 10th International Conference on Cloud Computing, Data Science and Engineering (Confluence).
Martzloff, J. C. (2008). Mathematics in Japan Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (pp. 1396-1400): Springer.
Mehra, J. and Rechenberg, H. (1982). The historical development of quantum theory. New York: Springer-Verlag.
Miller, G. A. (1930). On the History of Determinants. The American Mathematical Monthly, 37(5), 216-219. doi:10.2307/2299112
Mills, W. H., Robbins, D. P. and Rumsey, H. (1986). Self-complementary totally symmetric plane partitions. Journal of Combinatorial Theory, Series A, 42(2), 277-292. doi:http://dx.doi.org/10.1016/0097-3165(86)90098-1
Moler, C. (1974). Cramer's rule on 2-by-2 systems. ACM SIGNUM Newsletter, 9(4), 13-14.
Muir, T. (1881). The Law of Extensible Minors in Determinants. Transactions of the Royal Society of Edinburgh, 30(01), 1-4.
Muir, T. (1906). The theory of determinants in the historical order of development (Vol. 1): Macmillan and Company, limited.
Muir, T. (1911a). History of the Theory of Determinants (Vol. I-III). London: MacMillan.
Muir, T. (1911b). The Theory of Determinants in the Historical Order of Development (Vol. II): Macmillan and Company, limited.
Pan, V. Y., Yu, Y. and Stewart, C. (1997). Algebraic and numerical techniques for the computation of matrix determinants. Computers and Mathematics with Applications, 34(1), 43-70.
Pickover, C. A. (2011). A passion for mathematics: numbers, puzzles, madness, religion, and the quest for reality. John Wiley and Sons.
Rao, C. R. and Mitra, S. K. (1972). Generalized inverse of a matrix and its applications. In the Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. pp. 601-620.
Raptis, M., Kirovski, D. and Hoppe, H. (2011). Real-time classification of dance gestures from skeleton animation. In the Proceedings of the 2011 ACM SIGGRAPH/Eurographics symposium on computer animation.
Rezaifar, O. and Rezaee, H. (2007). A new approach for finding the determinant of matrices. Applied Mathematics and Computation, 188(2), 1445-1454. doi:10.1016/j.amc.2006.11.010
Rice, A. and Torrence, E. (2006). Lewis Carroll ’ s Condensation Method for Evaluating Determinants. Mathematics Association of America(November), 12-15.
Rice, A. and Torrence, E. (2007). Shutting up like a telescope": Lewis Carroll's" Curious Condensation Method for Evaluating Determinants. The college mathematics journal, 38(2), 85-95.
Robbins, D. and Rumsey, H. (1986). Determinants and alternating sign matrices. Advances in Mathematics, 62(2), 169-184. doi:10.1016/0001-8708(86)90099-X
Robbins, D. P. (2005). A conjecture about Dodgson condensation. Advances in Applied Mathematics, 34(4), 654-658.
Robinson, S. M. (1970). A short proof of Cramer's rule. Mathematics Magazine, 94-95.
Rohil, H. and Kaushik, P. (2014). Adjacency Matrix based Face Recognition Approach. International Journal of Computer Applications, 98(20).
Rothman, T. and Fukagawa, H. (1998). Japanese temple geometry. Scientific American, 278(5), 84-91.
Russell, B. and Whitehead, A. N. (1913). Principia mathematica to* 56. Cambridge UK: Cambridge University Press.
Saaty, T. L. (2003). Decision-making with the AHP: Why is the principal eigenvector necessary. European journal of operational research, 145(1), 85-91.
Salihu, A. (2012). New Method to Calculate Determinants of n × n ( n ≥ 3 ) Matrix , by Reducing Determinants to 2nd Order. International Journal of Algebra 6(19), 913-917.
Schmidt, A. D. and Greene, J. R. (2011). Dodgson’s Determinant: A Qualitative Analysis. Journal of Linear Algebra, 2(13), 34-54.
Searle, S. R. (2000). The infusion of matrices into statistics. IMAGE: Bulletin of international linear algebra society, 24, 25–32.
Shafarevich, I. and Remizov, A. (2013). Matrices and Determinants. Linear Algebra and Geometry SE - 2, 25-77 LA - English.
Shallit, J. (1994). Origins of the analysis of the Euclidean algorithm. Historia Mathematica, 21(4), 401-419. doi:https://doi.org/10.1006/hmat.1994.1031
Shen, K., Crossley, J. N., Lun, A. W. C. and Liu, H. (1999). The nine chapters on the mathematical art: Companion and commentary. Beijing: Oxford University Press.
Shiflet, A. B. and Shiflet, G. W. (2011). Introducing Matrix Operations through Biological Applications. Journal Of Computational Science, 2(1), 15-20.
Shores, T. S. (2007). Applied linear algebra and matrix analysis: Springer Science and Business Media.
Sobamowo, M. (2016). On the Extension of Sarrus’ Rule to Matrices: Development of New Method for the Computation of the Determinant of Matrix. International Journal of Engineering Mathematics, 2016.
Sobczyk, G. (2002). Generalized Vandermonde determinants and applications. Aportaciones Matematicas, Serie Comunicaciones, 30, 203-213.
Stocco, L. J., Salcudean, S. E. and Sassani, F. (1999). On the use of scaling matrices for task-specific robot design. IEEE Transactions on Robotics and Automation, 15(5), 958-965.
Studnička, F. J. (1876). AL Cauchy als formaler begründer der determinanten-theorie: Eine literarische-historische studie (Vol. 8): Verl der Königl. Böhmischen Gesellschaft der Wissenschaften.
Sueur, J., Aubin, T. and Simonis, C. (2008). Equipment review: seewave, a free modular tool for sound analysis and synthesis. Bioacoustics, 18(2), 213-226.
Sylvester, J. J. (1867). Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(232), 461-475.
Sylvester, J. J. and Baker, H. F. (2012). The collected mathematical papers of James Joseph Sylvester (Vol. 3): Cambridge University Press.
Taheri, S. M., Boostanpour, J. and Mohammadi, B. (2013). A Novel Algorithm for determinant calculation of N× N matrix. In the Proceedings of the 2013 International Conference on Advances in Computing, Communications and Informatics (ICACCI).
Tarokh, V., Seshadri, N. and Calderbank, A. R. (1998). Space-time codes for high data rate wireless communication: Performance criterion and code construction. Information Theory, IEEE Transactions on, 44(2), 744-765.
Tarski, A. (1946). Introduction to Logic and the Methodology of Deductive Sciences. Dover Publication, Inc.
Tirkkonen, O. and Hottinen, A. (2002). Square-matrix embeddable space-time block codes for complex signal constellations. Information Theory, IEEE Transactions on, 48(2), 384-395.
Tucker, A. (1993). The growing importance of linear algebra in undergraduate mathematics. The college mathematics journal, 24(1), 3-9.
Tweedie, C. (1915). A study of the life and writings of Colin MacLaurin. The mathematical gazette, 8(119), 133-151.
Ufuoma, O. (2013). A New and Simple Method of Solving Large Linear Systems : Based on Cramer’ s Rule but Employing Dodgson’ s Condensation. In the Proceedings of the World Congress on Engineering and Computer Science, San Francisco.
Urbańska, A. (2008). Faster Combinatorial Algorithms for Determinant and Pfaffian. Algorithmica, 56(1), 35-50. doi:10.1007/s00453-008-9240-9
Vaishnav, C., Choucri, N. and Clark, D. (2013). Cyber international relations as an integrated system. Environment Systems and Decisions, 33(4), 561-576.
Vandermonde, A. T. (1772). Mémoire sur l’élimination. Mémoires de l’Académie Paris, II, 516-532.
Vein, R. and Dale, P. (1999). Determinants and their applications in mathematical physics.. Springer Science and Business Media.
Vysotskaya, A. (2018). Accounting Games: Using Matrix Algebra in Creating the Accounting Models. Mathematics, 6(9), 152-160.
Watkins, D. S. (2004). Fundamentals of matrix computations (Vol. 64). John Wiley and Sons.
Weber, H. J. and Arfken, G. B. (2003). Essential mathematical methods for physicists. ISE: Elsevier.
Wedderburn, J. H. M. (1934). Lectures on matrices (Vol. 17). American Mathematical Soc.
Weld, L. G. (1893). A short course in the theory of determinants. Macmillan and Company.
Published
2022-07-02
How to Cite
BabarinsaO., SofiA. Z. M., MohdA. H., EluwoleA., SundayI., Adamu W., OnojhojobiB., Sheidu M., KehindeR., Daniel L., DolapoS., EdogbanyaH. O., Oyem A., AdeniyiI., EzenwekeC., UmaruS., Emmanuel S., BitrusK., Cyril-Okeme,V., UpahiE., AkaligwoE. A., Arinze L., BargumaS., EdiboF., AttehM., IdokoA., OpeyemiE., Adesanmi M., OjoO., DisuA., Adeeko T., AnukwuG., SamsonD., OgunleyeK., KoffaJ., James ojonubah, MusaM., ChojiN. M., EmekaH. O., UmarA., HassanM., JaiyeobaO., AidaraS., MammanM., MangaE., SuleA., UyimwenO., AhmedA.- al-M., GarbaZ., GbeneZ., JubriS., & OkekeG. (2022). NOTE ON THE HISTORY OF (SQUARE) MATRIX AND DETERMINANT. FUDMA JOURNAL OF SCIENCES, 6(3), 177 - 190. https://doi.org/10.33003/fjs-2022-0603-775