The Determinant from Signed Volume to the Laplace Expansion

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Abstract

We present a novel perspective on developing the determinant through the lens of signed volume. Starting with a unique and rigorous development of both the volume and orientation of a parallelepiped, we are able to give an unambiguous, basis-free definition for the determinant of a linear transformation. We then build intuition for the determinant by proving many of its properties in a succinct and basis-free fashion. We conclude our journey by using these properties to derive a well-known method for computing the determinant and motivating the Laplace expansion.

Original languageEnglish (US)
Pages (from-to)437-447
Number of pages11
JournalAmerican Mathematical Monthly
Volume126
Issue number5
DOIs
StatePublished - May 28 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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