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Show matrix has no inverse

  1. Apr 19, 2008 #1
    [SOLVED] Show matrix has no inverse

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

    Show that for a matrix A that has dimensions n by n (where n is an odd number) and is skew symmetric (i.e. transpose of A = -A) that it has no inverse

    3. The attempt at a solution

    Not sure where to start but have a feeling I have to used the definition transpose of A = -A to find the determinant first.
     
  2. jcsd
  3. Apr 19, 2008 #2

    cristo

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    Good idea. If you can relate the determinant of A to the determinant of -A, then you may have a chance.
     
  4. Apr 19, 2008 #3
    For starters, I've decided to choose n=3. When creating matrix A and hence -A, should I just nominate values of a,b,c etc and have the determinant in terms of each matrix entry like I've shown below? Where to from here?

    [​IMG]
     
    Last edited: Apr 19, 2008
  5. Apr 19, 2008 #4

    cristo

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    No. You should keep n as an arbitrary odd number, and keep A an arbitrary matrix, if you want to answer the question.

    Hint: How are the determinant of a matrix and its transpose related?
     
  6. Apr 19, 2008 #5
    try to look what is
    [tex]A^T A [/tex]
    then use the rule for determinant of product.
     
  7. Apr 19, 2008 #6
    uhh sorry, just look what determinant of transpose matrix is as cristo said.
     
  8. Apr 19, 2008 #7
    [​IMG]

    What then? :confused:
     
  9. Apr 19, 2008 #8

    Hootenanny

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    Correct. You have yet to make use if this information,
     
  10. Apr 19, 2008 #9
    Ok, so
    |A| = |transpose of A| = |-A|

    But how do I show no inverse exists?
     
  11. Apr 19, 2008 #10
    |-A|=(-1)^n|A|
    but you need to know this before using this.
    and there is no other way here.
     
  12. Apr 19, 2008 #11

    cristo

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    You should not give solutions to homework problems (and this is, in essence, a solution).

    You can give hints, or suggestions of ways to attack the problem. For example, one could ask if the OP could express det(pA) in terms of det(A), in the case of an nxn matrix. The result then follows on setting p=-1. However, just giving a formula does not help the student learn anything.
     
  13. Apr 19, 2008 #12
    so now
    |A| = |transpose of A| = |-A| = (-1)^n|A|

    I still don't get it
     
  14. Apr 19, 2008 #13

    Hootenanny

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    t_n_p, as cristo has said, you have virtually been given the answer by LQG. Try thinking about what has been said for a while before asking further questions. We won't hold you're hand through every single step of every single question, you have to do some work.
     
  15. Apr 19, 2008 #14
    I've thought about it. The formula provided by LQG does not register anything to me. Yes, I know
    [​IMG]
    and the determinant of A can supposedly be found using the formula provided by LQG, but my denominator thus appears merely as (-1)^n|A|. I am also unfamiliar with raising constants to the power of a matrix. I can not see how this is relevant or useful to proving the inverse does not exist. Hence my question is how is it related to what I want to know?
     
  16. Apr 19, 2008 #15

    cristo

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    What is the condition for the existence of the inverse to a matrix, in terms of the determinant? This is something you learn in the first few weeks of a linear algebra course, so if you're not sure, then look through your course notes, or textbook.
     
  17. Apr 19, 2008 #16
    yes, but from his posts here you can see that he doesnt recall this equation, this is why i urged him to prove it and then use it, if he just uses then he will not learn anything.
     
  18. Apr 19, 2008 #17
    for the inverse to exist, determinant must not equal zero. So if I set the equation given by LQG and equate to zero, the inverse will not exist?

    i.e. (-1)^n|A| = 0
     
    Last edited: Apr 19, 2008
  19. Apr 19, 2008 #18

    Hootenanny

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    Correct. You have a further condition on the matrix which you have not yet used.
     
  20. Apr 19, 2008 #19

    cristo

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    No, now's the time you need to think a little. Why are you setting that to zero? It seems like you're just doing it to satisfy the condition!!

    You know that, since the matrix is skew-symmetric, that det(A)=det(-A). You also have the expression that lqg gave you: det(-A)=(-1)^n.det(A). Put these two together, and what do you get?

    I'm not sure how I can help any more without explicitly telling you the answer!
     
  21. Apr 19, 2008 #20
    Ok, I think I'm looking too hard into this.

    Basically I want to show that |A| = 0, and hence the inverse does not exist.

    I know that
    [​IMG]

    Say I ignore |-A| for the time being, and therefore I have |A| = (-1)^n|A|. I then divide both sides by |A| and take log10 both sides. Am I on the right track?
     
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