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Simplify the matrix product to the identity

  1. Nov 16, 2014 #1
    1. The problem statement, all variables and given/known data
    IF G, H and G+H are invertible matrices and have the same dimensions
    Prove that G(G^-1 + H^-1)H(G+H)^-1 = I

    3. Attempt
    G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1
    = (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1
    =2GH^-1+ GH^-1GH^-1

    I am not sure where to go from here.
     
  2. jcsd
  3. Nov 16, 2014 #2

    LCKurtz

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    Try starting with multiplying out what I have highlighted in red, then multiply the blue into that and see what it looks like.
     
    Last edited by a moderator: Nov 17, 2014
  4. Nov 16, 2014 #3

    ehild

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    Where did that starting expression come from????
    Just expand the multiplications from left to right. What is G(G-1+H-1)?
    Multiply the result with H from the right . What do you get?

    LCKurtz was faster... :D
     
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