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Linear Algebra, Optimisation and Numerical Methods / Determinants

The Determinant: Defining Properties

Fact 2 of the inverse-matrices note flagged ad−bcad-bc as the exact quantity separating invertible from singular 2×22\times2 matrices, without explaining where that formula comes from or how it generalises. This note answers both: it defines the determinant not by a formula but by three properties, shows the properties alone force the familiar 2×22\times2 and 3×33\times3 formulas, and states the single fact this unit needs most: det⁡A=0\det A = 0 exactly when AA is singular.
Learning objectives. By the end of this note you should be able to:
  • state the three defining properties of the determinant;
  • derive the 2×22\times2 formula from the three properties alone;
  • verify det⁡I=1\det I=1, the sign-flip on row exchange, and multilinearity on small examples;
  • use det⁡A=0\det A=0 as an invertibility test.

In this lesson

  1. The three defining properties
  2. Deriving the 2×22\times2 formula
  3. Determinant and invertibility

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