Obtaining inverses of block matrices

  • Thread starter s3a
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  • #1
s3a
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Homework Statement


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}}

Homework Equations


N/A

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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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