# Basic Properties of Determinants of Square Matrices over a Field 1

Formalized Mathematics (2007)

• Volume: 15, Issue: 1, page 17-25
• ISSN: 1426-2630

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## Abstract

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In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement [...] where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with a repeated line. Finally, at the end I prove that the determinant of a product of two square matrices is equal to the product of their determinants.MML identifier: MATRIX11, version: 7.8.04 4.81.962

## How to cite

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Karol Pąk. " Basic Properties of Determinants of Square Matrices over a Field 1 ." Formalized Mathematics 15.1 (2007): 17-25. <http://eudml.org/doc/267525>.

@article{KarolPąk2007,
abstract = {In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement [...] where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with a repeated line. Finally, at the end I prove that the determinant of a product of two square matrices is equal to the product of their determinants.MML identifier: MATRIX11, version: 7.8.04 4.81.962},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {1},
pages = {17-25},
title = { Basic Properties of Determinants of Square Matrices over a Field 1 },
url = {http://eudml.org/doc/267525},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Karol Pąk
TI - Basic Properties of Determinants of Square Matrices over a Field 1
JO - Formalized Mathematics
PY - 2007
VL - 15
IS - 1
SP - 17
EP - 25
AB - In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement [...] where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with a repeated line. Finally, at the end I prove that the determinant of a product of two square matrices is equal to the product of their determinants.MML identifier: MATRIX11, version: 7.8.04 4.81.962
LA - eng
UR - http://eudml.org/doc/267525
ER -

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