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Linear algebra

  1. Mar 27, 2010 #1
    1. The problem statement, all variables and given/known data

    a)Let A be an n*n invertible matrix. Show that the inverse of the transpose of a matrix (A^{T}) is (A⁻¹)^{T}

    b)Let A,B, 2A+B be n×n matrices.Show that A⁻¹+2B⁻¹ is also invertible and express (A⁻¹+2B⁻¹)⁻¹ in terms of A, B and (2A+B)⁻¹

    2. Relevant equations

    I have done part a. I have been stuck on part b for a while. I have tried everything i can think of.



    3. The attempt at a solution

    I have tried revamping
    (A⁻¹+2B⁻¹)⁻¹(A⁻¹+2B⁻¹)=I
    (A⁻¹+2B⁻¹)⁻¹A⁻¹+ (A⁻¹+2B⁻¹)⁻¹2B⁻¹=I
    (A⁻¹+2B⁻¹)⁻¹=(I-(A⁻¹+2B⁻¹)⁻¹2B⁻¹)A
    Then simplified it, but the result is not in terms of the matrix (2A+B)⁻¹,A and B.
    This question really has me stumped.
     
  2. jcsd
  3. Mar 27, 2010 #2

    tiny-tim

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    Hi oddiseas! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    Hint: start with the result :wink:, and see what values of p q and r make it work …

    multiply (A-1 + 2B-1) by pA + qB + r(2A + B)-1

    what do you get? :smile:
     
  4. Mar 27, 2010 #3
    I have multiplied (A⁻¹+B⁻¹) by what you have suggested and i get:

    p+2q+qBA⁻¹+2pAB⁻¹+r(2A²+BA)⁻¹+r(2B²+4AB)⁻¹

    But i still cannot see the logic. This question just seems to get longer and longer!
    Could you explain to me the logic in how i should approach this question.
     
  5. Mar 27, 2010 #4

    Dick

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    It said (2B^(-1)+A^(-1))^(-1) is a combination of the given matrices. It didn't say it was a linear combination. You just have to fool around with combinations. Hint: what is A^(-1)*(2A+B)*B^(-1)?
     
  6. Mar 27, 2010 #5
    Thanks for the reply, that helps a lot. Your'e a superstar! I will work on that and see what i can get, and then post it.
     
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