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Commutative matrices

  1. Mar 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Proof that if two matrices A and B are commutative, BA=AB, then the equations:

    a) [tex](A+B)^2 = A^2 + 2AB + B^2[/tex] ; b)[tex](A+B)^3=A^3+3A^2B+3AB^2+B^3[/tex]

    are true.

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Mar 23, 2008
  2. jcsd
  3. Mar 23, 2008 #2

    cristo

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    Have you tried anything? What do you get if you explicitly expand out the brackets in each case? What happens in the case of A and B commuting?
     
  4. Mar 23, 2008 #3
    I don't know what do you mean?
     
  5. Mar 23, 2008 #4

    cristo

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    What do you get when you multiply out (A+B)(A+B), where A and B are matrices?
     
  6. Mar 23, 2008 #5
    I get
    [tex](A+B)(A+B)=A^2+AB+BA+B^2[/tex]
     
  7. Mar 23, 2008 #6

    cristo

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    Good, now can you simplify this in the case that A and B commute?
     
  8. Mar 23, 2008 #7

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

    like this?
     
  9. Mar 23, 2008 #8

    cristo

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    Correct!

    Now, try applying similar techniques to the second question.
     
  10. Mar 23, 2008 #9
    [tex](A+B)^3=(A+B)(A+B)(A+B)=(A^2+2AB+B^2)(A+B)=A^3+A^2B+2A^2B+2AB^2+B^2A+B^3=A^3+A*AB+2A*AB+2

    AB*B+B*BA+B^3[/tex]
    [tex]=A^3+3A^2B+3AB^2+B^3[/tex]

    Something like this?
     
    Last edited: Mar 23, 2008
  11. Mar 24, 2008 #10
    *Applause*

    Huzza, well done!
     
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