Abstract
In the second chapter we deal with matrices and determinants. The chapter starts with determinants of second and third orders, which are defined through solutions of linear algebraic systems; determinants of arbitrary order are defined inductively. The basic properties of determinants are investigated. We then take a look at determinants from a more abstract viewpoint: it is proved that the determinant of a square matrix can be defined as an antisymmetric multilinear function of the rows. Using some basic elements of permutation theory, we continue to study the properties of determinants; in particular, we derive explicit formula for determinants. Finally, we define the rank of a matrix and the main operations on matrices (sum, product, inverse matrix) and investigate their properties.
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Notes
- 1.
We are being a bit sloppy with language here. We have defined the determinant as a function that assigns a number to a matrix, so when we speak of the “rows of a determinant,” this is shorthand for the rows of the underlying matrix.
- 2.
For the definition and a discussion of antisymmetric functions, see Sect. 2.6.
- 3.
For example, photons are bosons, and the particles that make up the atom—electrons, protons, and neutrons—are fermions.
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© 2012 Springer-Verlag Berlin Heidelberg
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Shafarevich, I.R., Remizov, A.O. (2012). Matrices and Determinants. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_2
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DOI: https://doi.org/10.1007/978-3-642-30994-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30993-9
Online ISBN: 978-3-642-30994-6
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