Demystifying the determinant of a matrix

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.