Demystifying the determinant of a matrix

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 d e t A, for which the next properties holds: 1. If B is a matrix obtained from A exchanging two rows (or columns) then d e t B = d e t A; 2. If one of the rows (or columns) of A is a linear combination of the other then d e t A = 0; 3. d e t I = 1, where I is the identity matrix. If the matrix A has order n = 1 then d e t A = a 1 1. In the case n = 2, d e t A = a 1 1 a 2 2 a 1 2 a 2 1 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 d e t I = 1 and, moreover, show that this function coincides with the usual 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 de fi nition abstract, verifying that this one is simpler than the other.