Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Determinant of row interchange proof

  1. Oct 14, 2012 #1
    Need a lot of help here guys.

    I need to prove that for an nxn matrix A, if i interchange two rows to obtain B, then det=-detA

    I have proved my basis (below), but i'm stuck on the hard part, the induction (which i'm required to do). I understand the steps of induction, but i don't know how to do it for this case.

    What i have so far:

    Let A be an nxn matrix.
    Basis n=2
    Then detA=a(11)a(22) - a(12)a(21)
    Now let B be the matrix obtained by interchanging rows 1 and 2
    Then detB=a(21)a(12) - a(22)a(11)
    =-detA
    So true for an arbitary 2x2 matrix.

    (induction)
    Assume true for n=k
    For a kxk matrix...?????
     
  2. jcsd
  3. Oct 15, 2012 #2

    chiro

    User Avatar
    Science Advisor

    Hey fackert.

    Try introducing a pre-multiplication matrix that swaps the rows and then use the relationship that det(A*B) = det(A)*det(B) where B is your original matrix and A is a matrix transformation that swaps the rows.
     
  4. Oct 15, 2012 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    my notes on determinants are on page 62ff of these notes:

    http://www.math.uga.edu/%7Eroy/4050sum08.pdf [Broken]
     
    Last edited by a moderator: May 6, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook