Properties of determinants

  1. Properties of Determinants Overview:
  2. Row Operation Property
    • Adding/Subtracting Rows → det B=B= det AA
    • Interchanging rows → det B=B=- det AA
    • Multiplying rows → det B=k  B=k\; \cdot det AA
    • Triangular Matrices
  3. The Multiplicative Property/Other Properties
    • det (A)=(A)= det (AT)(A^T)
    AA is invertible if and only if det A0A \neq 0
    • det (AB)=(AB)= det A  A \; \cdot det BB
  4. Applications to Determinants
    • If det (A)0(A) \neq 0, then columns are linearly independent
    • If det (A)=0(A) = 0, then columns are linear dependent
  1. Calculating the Determinant
    Compute det AA by row reduction to echelon form, where Compute det A by row reduction to echelon form.
    1. Find the determinants where Find the determinants,
      1. Find the determinants
      2. Find the determinants
    2. Property/Applications of Determinants
      You are given that matrix A, Property/Applications of Determinants and matrix A, Property/Applications of Determinants.
      1. Show that det (A)=(A)= det (AT)(A^T).
      2. Show that det (AB)=(AB)= det A  A \; \cdot det BB.
    3. Use determinants to decide if the set of vectors are linearly independent
      are the set of vectors linearly independent
      1. Proving the Property of Determinants
        Show that if a square matrix AA invertible, then
        det(A1)=1det(A)\det (A^{-1})=\frac{1}{\det (A)}
        1. Show that if 2 rows of a square matrix AA are the same, then det A=0A=0