International Journal of Engineering Mathematics

Volume 2016 (2016), Article ID 9382739, 14 pages

http://dx.doi.org/10.1155/2016/9382739

Research Article

## On the Extension of Sarrus’ Rule to Matrices: Development of New Method for the Computation of the Determinant of Matrix

Department of Mechanical Engineering, University of Lagos, Lagos, Nigeria

Received 14 June 2016; Revised 8 August 2016; Accepted 30 August 2016

Academic Editor: Giuseppe Carbone

Copyright © 2016 M. G. Sobamowo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- O. Rezaifar and H. Rezaee, “A new approach for finding the determinant of matrices,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1445–1454, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Abbott, J. Bronstein, and M. Mulders, “Fast deterministic computation of determinants of dense matrices,” in
*Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '99)*, S. Dooley, Ed., pp. 197–204, ACM, Vancouver, Canada, July 1999. View at Publisher · View at Google Scholar - A. A. M. Ahmed and K. L. Bondar, “Modern method to compute the determinants of matrices of order 3,”
*Journal of Informatics and Mathematical Sciences*, vol. 6, no. 2, pp. 55–60, 2014. View at Google Scholar - K. L. Clarkson, “Safe and effective determinant evaluation,” in
*Proceedings of the 33rd Annual Symposium on Foundations Computer Science*, pp. 387–395, IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, IEEE Computer Press, The Institute of Electrical and Electronics Engineers, Pittsburgh, Pa, USA, 1992. - B. M. Dingle, “Calculating determinants of symbolic and numeric matrices,” Texas, 2005.
- C. Dubbs and D. Siegel, “Computing determinants,”
*The College Mathematics Journal*, vol. 18, no. 1, pp. 48–50, 1987. View at Publisher · View at Google Scholar - D. Eberly,
*The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices*, Geometric Tools, LLC, Scottsdale, Ariz, USA, 2007. - W. M. Gentleman and S. C. Johnson, “The evaluation of determinants by expansion by minors and the general problem of substitution,”
*Mathematics of Computation*, vol. 28, no. 26, pp. 543–548, 1974. View at Google Scholar · View at MathSciNet - A. Salihu and Q. Gjonbalaj, “New method to compute the determinant of a 4x4 matrix,” in
*Proceedings of the 3rd International Mathematics Conference on Algebra and Functional Analysis*, At Elbasan, Albania, May 2009. - A. Assen and J. Venkateswara Rao, “A study on the computation of the determinants of a $3\times 3$ matrix,”
*International Journal of Science and Research*, vol. 3, no. 6, pp. 912–921, 2014. View at Google Scholar - F. Chió,
*Mémoire sur les Fonctions Connues Sous le Nom de Résultantes ou de Déterminants*, E. Pons, Turin, Italy, 1853. - C. L. Dodgson, “Condensation of determinants, being a new and brief method for computing their arithmetic values,”
*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*, vol. 15, pp. 150–155, 1866. View at Publisher · View at Google Scholar - C. L. Dodgson,
*Elementary Treatise on Determinants with the Applications to Simultaneous Linear Equations and Algebraical Geometry*, MacMillan, London, UK, 1867. - M. E. A. El-Mikkawy, “A fast algorithm for evaluating nth order tri-diagonal determinants,”
*Journal of Computational and Applied Mathematics*, vol. 166, no. 2, pp. 581–584, 2004. View at Publisher · View at Google Scholar · View at Scopus - E. K. Tofen and G. Villard, “On the complexity of computing determinants
^{∗}(extended abstract),” in*Proceedings of the 5th Asian Symposium on Computer Mathematics (ASCM '01)*, K. Shirayanagi and K. Yokoyama, Eds., vol. 9 of*Lecture Notes Series on Computing*, pp. 13–27, World Scientific, Singapore, 2001. - W. M. Gentleman and S. C. Johnson, “Analysis of algorithms, a case study: determinants of polynomials,” in
*Proceedings of the 5th annual ACM Symposium on Theory of Computing*, pp. 135–141, ACM Press, 1973. - Q. Gjonbalaj and A. Salihu, “Computing the determinants by reducing the orders by four,”
*Applied Mathematics E-Notes*, vol. 10, pp. 151–158, 2010. View at Google Scholar · View at MathSciNet - D. Hajrizaj, “New method to compute the determinant of $3\times 3$ matrix,”
*International Journal of Algebra*, vol. 3, no. 5, pp. 211–219, 2009. View at Google Scholar - D. Henrion and M. Šebek, “Improved polynomial matrix determinant computation,”
*IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, vol. 46, no. 10, pp. 1307–1308, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C. F. Ipsen and D. J. Lee, “Determinant approximations,” in
*Numerical Linear Algebra with Applications*, John Wiley & Sons, New York, NY, USA, 2005. View at Google Scholar - E. Kaltofen, “On computing determinants of matrices without divisions,” in
*Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '92)*, P. S. Wang, Ed., pp. 342–349, ACM, 1992. View at Publisher · View at Google Scholar - E. Kaltofen and G. Villard, “Computing the sign or the value of the determinant of an integer matrix, a complexity survey,”
*Journal of Computational and Applied Mathematics*, vol. 162, no. 1, pp. 133–146, 2001. View at Google Scholar - L. G. Molinari, “Determinants of block tridiagonal matrices,”
*Linear Algebra and Its Applications*, vol. 429, no. 8-9, pp. 2221–2226, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - V. Y. Pan, “Computing the determinant and the characteristic polynomial of a matrix via solving linear systems of equations,”
*Information Processing Letters*, vol. 28, no. 2, pp. 71–75, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Radić, “A generalization of the determinant of a square matrix and some of its applications in geometry,”
*Matematika*, vol. 20, pp. 19–36, 1999 (Serbo-Croatian). - R. Adrian and E. Torrence, “Shuttling up like a telescope': lewis carroll's ‘curious’ condensation method for evaluating determinants,”
*College Mathematics Journal*, vol. 38, no. 2, 2007. View at Google Scholar - H. Teimoori, M. Bayat, A. Amiri, and E. Sarijloo, “A new parallel algorithm for evaluating the determinant of a matrix of order n,”
*Euro Combinatory*, pp. 123–134, 2005. View at Google Scholar - X.-B. Chen, “A fast algorithm for computing the determinants of banded circulant matrices,”
*Applied Mathematics and Computation*, vol. 229, pp. 201–207, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Goldfinger, “Determinant by cofactor expansion using the cell processor,” CMSC 49lA, 2008.
- D. Bozkurt and T.-Y. Tam, “Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-lucas numbers,”
*Applied Mathematics and Computation*, vol. 219, no. 2, pp. 544–551, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Lang,
*Undergraduate Algebra*, Springer, New York, NY, USA, 2nd edition, 1990. - T. Sogabe, “A fast numerical algorithm for the determinant of a pentadiagonal matrix,”
*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 835–841, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X.-G. Lv, T.-Z. Huang, and J. Le, “A note on computing the inverse and the determinant of a pentadiagonal Toeplitz matrix,”
*Applied Mathematics and Computation*, vol. 206, no. 1, pp. 327–331, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - V. Pan, “Complexity of computations with matrices and polynomials,”
*SIAM Review*, vol. 34, no. 2, pp. 225–262, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - S. Lipschutz and M. Lipson,
*Schaum's Outlines Linear Algebra*, McGraw-Hill Companies, New York, NY, USA, 3rd edition, 2004. - J. Dutta and S. C. Pal, “Generalization of a new technique for finding the determinant of matrices,”
*Journal of Computer and Mathematical Sciences*, vol. 2, no. 2, pp. 266–273, 2011. View at Google Scholar - S.-Q. Shen, J.-M. Cen, and Y. Hao, “On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers,”
*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9790–9797, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Braae,
*Matrix Algebra: A Programmed Introduction*, John Wiley & Sons, New York, NY, USA, 1969. - M. C. Pease,
*Methods of Matrix Algebra*, Academic Press, New York, NY, USA, 1965. - C. H. Jepsen,
*The Matrix Algebra Calculator: Linear Algebra Problems for Computer Solution*, Brooks Cole, Pacific Grove, Calif, USA, 1988. - C. R. Rao,
*Matrix Algebra and its Applications to Statistical and Econometrics*, World Scientific, Singapore, 1998. - D. R. Hill,
*Modern Matrix Algebra*, Prentice-Hall, Upper Saddle River, NJ, USA, 2001. - J. R. Rice,
*Matrix Computations and Mathematical Software*, McGraw-Hill, 1985. View at MathSciNet