Demystifying the determinant of a matrix

Authors

  • Evandro Monteiro Instituto de Ciências ExatasUniversidade Federal de Alfenas

Keywords:

Determinant of a matrix, Laplace’s Theorem, multilinear function, Alternate function

Abstract

Determinant’s originated in the mid-seventeenth century when processes were studied to solve linear equations systems. The determinant of a matrix is a function that associate every square matrix A to a real number, denoted by detA, for which the next properties holds: 1. If B is a matrix obtained from A exchanging two rows (or columns) then detB = −detA; 2. If one of the rows (or columns) of A is a linear combination of the other then detA = 0; 3. detI = 1, where I is the identity matrix. If the matrix A has order n=1 then detA = a11. In the case n=2, detA = a11a22 − a12a21 and if n=3 is given by Sarrus Rule. For the calculation of the determinant of a matrix of order n>3 we use a more complicated procedure given by Laplace’s Theorem and as higher is the a order matrix as greater is the labor for calculation of it’s determinant. The objective of this paper is to present the determinant as a multilinear and alternate function such that detI = 1 and, moreover, show that this function coincides with theu sual determinant. We use concepts of Linear Algebra. We conclude this study comparing calculation of determinant of a matrix of order n>3 by Laplace’s Theorem and by this definition abstract, verifying that this one is simpler than the other.

 

Author Biography

Evandro Monteiro, Instituto de Ciências ExatasUniversidade Federal de Alfenas

Doutor em Estatística e Experimentação Agropecuária com Pós-doutorado em Estatística Multivariada

References

BUENO, H. P. Álgebra Linear: Um segundo curso. Rio de Janeiro: Editora SBM, 2006.

HOFFMAN, K.; KUNZE, R. Álgebra Linear. Tradução de Renate Watanabe. Segunda Edição. Rio de Janeiro: LTC, 1979.

LIMA, E. L. Álgebra Linear. Quarta ediçãoo. Coleção Matemática Universitária. Rio de Janeiro: IMPA, CNPQ, 2000.

ROSSO Jr., A. C.; FURTADO, P. Matemática: uma ciência para vida. Vol 2. Editora Harbra, 2011.

Published

31-12-2012

How to Cite

Monteiro, E. (2012). Demystifying the determinant of a matrix. Sigmae, 1(1), 33–43. Retrieved from https://publicacoes.unifal-mg.edu.br/revistas/index.php/sigmae/article/view/91