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Obtaining inverses of block matrices

  1. Feb 4, 2013 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Problem:
    Find and check the inverses (assuming they exist) of these three block matrices.:

    [1] {{I, 0},{C, I}}

    [2] {{A, 0}, {C, D}}

    [3] {{0, I}, {I, D}}

    Answer:
    [1] {{I, 0}, {-C, I}}

    [2] {{A^(-1), 0}, {-D^(-1) C A^(-1), D^(-1)}}

    [3] {{-D, I}, {I, 0}}

    2. Relevant equations
    N/A

    3. The attempt at a solution
    I have no idea how to begin answering this problem. Could someone please explain to me?

    If any more information is needed, just tell me and I will attempt to clarify the situation.

    Any help would be greatly appreciated!

    P.S.
    To read the matrices in the notation I used (which is Wolfram Alpha's notation), for [1], Row 1/Column 1 has I, Row 1/Column 2 has 0, Row 2/Column 1 has C, Row 2/Column 2 has I.
     
  2. jcsd
  3. Feb 4, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    It's pretty much just filling in the parts by step by step. Take the first one {{I, 0},{C, I}} and call the inverse {{W,X},{Y,Z}}. If you multiply the two together you want to get {{I,0},{0,I}}. The top left entry of the product matrix will be IW+0Y. You want that to be I. So W=I. Fill that in. Top right is IX+0Z. You want that to be 0. So you need to put X=0. Just keep going on like that.
     
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