@MISC{MATRIX11.MIZ,
  AUTHOR = {P\c{a}k, Karol},
  TITLE = {Basic Properties of Determinants of
           Square Matrices over a Field},
  SECTION1 = {The Sign of a Permutation},
  SECTION2 = {The Determinant of a Linear Combination of Lines},
  SECTION3 = {The Determinant of a Matrix with Permutated Lines
              and with a Repeated Line},
  SECTION4 = {The Determinant of a Product of Two Square Matrices},
  NOTE = {This work has been partially supported by the KBN grant
      4 T11C 039 24 and the FP6 IST grant TYPES No. 510996.},
  DAY = {21},
  MONTH = {March},
  YEAR = {2007},
  ADDRESS1 = {University of Bialystok, ul. Akademicka 2, 15-267 Bialystok},
  SUMMARY = {In this paper we present basic properties of the determinant of
      square matrices over a field and selected properties of the sign of a
      permutation.
      First, we define the sign of a permutation by the requirement
      $${\rm sgn}(p)=\prod_{1\leq i<j \leq n}\,{\rm sgn}(p(j)-p(i)),$$
      where $p$ is any fixed permutation of a set with $n$ elements.
      We 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 we 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.
      At the end, we prove that the determinant of a product of two square
      matrices is equal to the product of their determinants.}}

@BOOK{HerstenWinter,
        AUTHOR = {Herstein, I.N. and Winter, David J.},
         TITLE = {Matrix Theory and Linear Algebra},
     PUBLISHER = {Macmillan},
          YEAR = {1988}}

@BOOK{FriegInsel,
        AUTHOR = {Friedberg, Stephen H. and Insel, Arnold J. and Spence, Lawrence E.},
         TITLE = {Linear Algebra},
     PUBLISHER = {Prentice Hall},
          YEAR = {2002}}
