／var／logmarcus chiu

Determinants - Invertible／Non-Singular - Not-Invertible／Singular

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Determinants
  • is the scaling factor that a linear map/matrix has on volume
  • if the determinant of a square matrix 𝐴 is:
    • non-zero - then the matrix is invertible/non-singular (thus, there exists an inverse matrix 𝐴-1 such that 𝐴𝐴-1=𝐼)
    • zero - then the matrix is not-invertible/singular
  • non-square matrices do not have determinants

Determinant - Introduction

Determinant - Properties
  • 𝐷𝑒𝑡(𝐴) = 𝐷𝑒𝑡(𝐴𝑇)
  • 𝐷𝑒𝑡(𝐴𝐵) = 𝐷𝑒𝑡(𝐴)𝐷𝑒𝑡(𝐵)
  • 𝐷𝑒𝑡(𝐴𝑛) = 𝐷𝑒𝑡(𝐴)𝑛
  • 𝐷𝑒𝑡(𝐴-1) = 1/𝐷𝑒𝑡(𝐴)
  • The determinant of an upper or lower triangular matrix is the product of the diagonal entries

Determinant - Volume

Let 𝑣1, 𝑣2, …, 𝑣𝑛 be vectors in ℝ𝑛, let 𝑃 be the parallelepiped determined by these vectors, and let 𝐴 be the matrix with rows or columns 𝑣1, 𝑣2, …, 𝑣𝑛. Then the absolute value of the determinant of 𝐴 is the volume of 𝑃:

  • |𝐷𝑒𝑡(𝐴)| = volume of 𝑃

For proof expand below

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