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Can someone explain these 2 linear algebra proofs

  1. Jan 22, 2012 #1
    1. The problem statement, all variables and given/known data

    The proofs:

    show (A')^-1 = (A^-1)'

    and

    (AB)^-1 = B^-1A^-1


    2. Relevant equations



    3. The attempt at a solution

    for the first one:

    (A^-1*A) = I
    (A^-1*A)' = I' = I
    A'(A^-1)' = I

    but im not sure how this proves that a transpose inverse = a inverse transpose...

    the second i have the same problem. Not sure how this really proves what it's asking.

    http://tutorial.math.lamar.edu/Classes/LinAlg/InverseMatrices_files/eq0041M.gif [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jan 22, 2012 #2

    Mark44

    Staff: Mentor

    Both proofs are using the same basic idea: if the product of two (square) matrices is I, then the two matrices are inverses of each other.

    In the last line above, you have AT and (A-1)T multiplying to make I. That means that AT is the inverse of (A-1)T.
    Same thing in the work above, which says that the inverse of AB is B-1A-1.
     
    Last edited by a moderator: May 5, 2017
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