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Quick Linear Algebra Proof

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that if A is nonsingular then A^T is nonsingular and

    (A^T)^(-1) = (A^(-1))^T

    2. Relevant equations

    (AB)^T = (B^T)(A^T)



    3. The attempt at a solution

    Step 1: Multiply both sides by B^T

    B^T * (A^T)^(-1) = B^T * (A^(-1))^T

    B^T * (A^T)^(-1) = (A^(-1)*B)^T

    (A^(-1)*B)^T = (A^(-1)*B)^T

    I feel that my last step is wrong. Any suggestions would be helpful.

    I was also noticing that if I take just the right side of the equation and do this

    A^T*(A^(-1))^T = (A^(-1)*A)^T

    = I^T = I

    which suggests that the left side has to equal I if I multiply it by A^T as well so

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

    which makes sense just looking at it.

    Thanks!
     
    Last edited: Mar 2, 2009
  2. jcsd
  3. Mar 2, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Start from A*A^(-1)=I. Now take the transpose of both sides and use your relevant equation. Stare at it for a while and figure out what it means.
     
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