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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)
    So true for an arbitary 2x2 matrix.

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


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    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


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    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
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