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Determinants

  • Thomas Banchoff
  • John Wermer
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Let A be a linear transformation with matrix \( \left( {\begin{array}{*{20}{c}} a\\ c \end{array}\;\begin{array}{*{20}{c}} b\\ d \end{array}} \right) \). The quantity
$$ ad\; - \;bc $$
is called the determinant of the matrix \( \left( {\begin{array}{*{20}{c}} a\\ c \end{array}\;\begin{array}{*{20}{c}} b\\ d \end{array}} \right) \) and is denoted
$$ \left| {\begin{array}{*{20}{c}} a\\ c \end{array}\;\;\;\begin{array}{*{20}{c}} b\\ d \end{array}} \right|\;{\rm{.}} $$
(1)
Expressed in these terms, Theorem 2.4 states that A has an inverse if and only if \( \left| {\begin{array}{*{20}{c}} a\\ c \end{array}\;\;\;\begin{array}{*{20}{c}} b\\ d \end{array}} \right|\; \ne \;{\rm{0}}{\rm{.}} \) We shall see that the determinant gives us further information about the behavior of A.

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Thomas Banchoff
    • 1
  • John Wermer
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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