Properties of determinants

Properties of determinants

Lessons

Let AA be a n×nn \times n square matrix. Then
1) If a multiple of one row of matrix AA is added to another row to produce matrix BB, then det B=B= det AA
2) If two rows of AA are interchanged to produce matrix BB, then det B=B=- det AA
3) If one row of AA is multiplied by kk to produce matrix BB, then det B=kB=k \; \cdot det AA

If AA is an n×nn \times n matrix, then det(AT)=(A^T )= det AA

A square matrix AA is invertible if and only if det A0A \neq 0.

The Multiplicative Property
If AA and BB are n×nn \times n matrices, then det AB=AB= (det AA)(det BB)

It may be useful to know that the determinant of a triangular matrix is the product of the diagonal entries. For example,
determinant of a triangular matrix
  • 1.
    Properties of Determinants Overview:
    a)
    Row Operation Property
    • Adding/Subtracting Rows → det B=B= det AA
    • Interchanging rows → det B=B=- det AA
    • Multiplying rows → det B=kB=k\; \cdot det AA
    • Triangular Matrices

    b)
    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 AA \; \cdot det BB

    c)
    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


  • 2.
    Calculating the Determinant
    Compute det AA by row reduction to echelon form, where Compute det A by row reduction to echelon form.

  • 3.
    Find the determinants where Find the determinants,
    a)
    Find the determinants

    b)
    Find the determinants


  • 4.
    Property/Applications of Determinants
    You are given that matrix A, Property/Applications of Determinants and matrix A, Property/Applications of Determinants.
    a)
    Show that det (A)=(A)= det (AT)(A^T).

    b)
    Show that det (AB)=(AB)= det AA \; \cdot det BB.


  • 5.
    Use determinants to decide if the set of vectors are linearly independent
    are the set of vectors linearly independent

  • 6.
    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)}

  • 7.
    Show that if 2 rows of a square matrix AA are the same, then det A=0A=0