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2. Some exercises

  1. May 17, 2005 #1
    B is a n*n matrix

    1. Let B^2 =B. Prove that either det(B) =1 or B is singular.
    2. If Transpose(B) = B^-1 , what is det(B)?


    Sorry I am asking, but I can't figure them out! I'd really like to improve my linear algebra skills.
    Thanks!
     
  2. jcsd
  3. May 17, 2005 #2

    dextercioby

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    The first is really easy.

    HINT:What is [itex] \mbox{det} \ A\cdot B [/itex] equal to...?(A,B matrices n*n).Then take A=B and recover the result u were supposed to prove.

    For the second HINT:multiply to the left by B and use the "det" property u used for 1.

    Daniel.
     
  4. May 17, 2005 #3
    I don't understand, more depth please?
     
  5. May 17, 2005 #4

    dextercioby

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    Alright.Take two n*n matrices A and B.Each of them has a determinant.Question:what is the product of their determinants...?

    [tex] \mbox{det} \ A\cdot B =...? [/tex]

    Daniel.
     
  6. May 17, 2005 #5
    det(a*b) = det(a)det(b)

    So for #1, I'm still stuck.
    What do I do?
     
  7. May 17, 2005 #6

    mathwonk

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    the point is that every matrix satisfies a polynomial, and that polynomial tells you rthe eigenvaleus, which tell you whetehr it is singualr or not.


    now a polynomial satisfied by a matrix such that A^2 = A would be????
     
  8. May 17, 2005 #7

    dextercioby

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    Perfect,then u must agree that

    [tex] \mbox{det} \ B^{2}=\left(\mbox{det} \ B\right)^{2} [/tex] (1)

    And now apply "det" on the equation

    [tex] B^{2}=B [/tex](2)

    and use (1) to get a quadratic algebraic eq. in [itex] \mbox{det} \ B [/itex].


    Daniel.
     
  9. May 17, 2005 #8
    dextercioby, I didn't quite get your each of your steps in the last post.

    Could you help me in writing the full proof for 1 and 2?
    Aaargh! I feel so frustrated. I should have take regular linear algebra instead of the honors one. I suck at proofs.


    I'm really sorry, but I need to solve these problems, but can't get them. Thanks!
     
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