Matrices/Linear Algebra Proof

  1. 1. The problem statement, all variables and given/known data

    Let A be an n x n matrix. Show that if A2=0 then I - A is non-singular and (I - A)-1= I + A

    2. Relevant equations



    3. The attempt at a solution

    Ok, so I started off with finding the general form of a 2x2 matrix which when squared gives a zero matrix, and all the properties above are satisfied. But how do I show that for an n x n matrix? Please help me out.
     
  2. jcsd
  3. Dick

    Dick 25,887
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    What is (I-A)*(I+A)?
     
  4. The Identity matrix.
     
  5. Dick

    Dick 25,887
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    Hence? What do you conclude from that?
     
  6. But I only know that because the question says to show it and the "experiment" I did with my 2x2 matrix shows that. I cannot prove it. Please give a more thorough explanation :)
     
  7. Dick

    Dick 25,887
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    If A*B=I then A=B^(-1). Isn't that the definition of inverse?
     
  8. Yes, obviously. I do not know how that is connected to the question though. It asks us to show that the inverse of I-A is I+A which at this point is not self-evident (or at least I think so).
     
  9. Dick

    Dick 25,887
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    Now you are just plain confusing me. If (I-A)*(I+A)=I then as I read the definition of 'inverse' that means (I-A)^(-1)=(I+A). If you mean showing its the inverse by using the explicit calculation you probably used for 2x2 matrices, then you can't do that. You don't know what A is. It's still true though.
     
  10. How? This is my question.
     
  11. Dick

    Dick 25,887
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    Multiply it out. (I+A)*(I-A)=? Use the distributive property! I thought you did that in post 3. I may have been wrong.
     
  12. Can you do that with matrices? I have read the property that (A+B)C=AC+BC but only to that extent.
     
  13. Dick

    Dick 25,887
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    Sure you can. (I+A)(I-A)=I(I-A)+A(I-A). Now distribute again.
     
  14. Yes, suppose C = D+E, hence AC+BC=(A+B)(D+E).
     
  15. Of course. That was kind of a stupid question. Thanks for clearing it up.

    The question first asks us to show that I-A is non-singular. How do we show that? Or does that follow from (I-A)(I+A)=I and since I has an inverse, it must be non-singular?
     
  16. When two matrices are multiplied together to produce the identity, it means the two matrices are inverses of each other. What is the definition of a non-singular matrix?
     
  17. Dick

    Dick 25,887
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    You KNOW I has an inverse. It's I! It's doesn't follow from that. Please quote me the definition of what an inverse is?
     
  18. A matrix multiplied by the original matrix to give Identity.
     
  19. Dick

    Dick 25,887
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    Ok, so (I+A)(I-A)=I. What's the inverse of (I-A)?
     
  20. I get that. That is what I'm asking. Is the proof of (I-A)'s non-singularity the fact that I+A is its inverse?
     
  21. Dick

    Dick 25,887
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    Yes, yes, yes. If a matrix has an inverse, it's nonsingular. Now I'm going to have to ask to you look up the definition of 'nonsingular'.
     
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