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Matrix problem about AB = BA

  1. Nov 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Let [tex]A, B \epsilon R[/tex]n x n.

    2. Relevant equations

    A. Show that if AB = BA, then
    (A + B)2 = A2 + 2AB + B2.

    B. Give an example of 2 x 2 matrices A and B such that
    (A + B)2 [tex]\neq[/tex] A2 + 2AB + B2.

    3. The attempt at a solution
    I have tried to find such a matrices A and B such that the requirements applies. Perhaps, this allows to show the equations.
     
    Last edited: Nov 9, 2008
  2. jcsd
  3. Nov 9, 2008 #2

    gabbagabbahey

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    For part (a), just expand [itex](A+B)^2[/itex]...what do you get when you do that?
     
  4. Nov 9, 2008 #3
    [tex]A^2 +2AB + B^2[/tex]. Do you suggest that this is enough for A?

    I accidentally managed to solve B: an example is A= <1, 0; 1, 1> and B= <1, 1; 0, 1>.
     
    Last edited: Nov 9, 2008
  5. Nov 9, 2008 #4

    gabbagabbahey

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    this is only true when AB=BA, since

    [tex](A+B)^2=(A+B)(A+B)=AA+AB+BA+BB=A^2+AB+BA+B^2[/tex]

    ...do you follow?
     
  6. Nov 9, 2008 #5
    Good Point! This must be enough for A. Thanks!
     
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