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Matrix Distributive Law

  1. Apr 11, 2015 #1
    1. The problem statement, all variables and given/known data
    Not really a homework question. Something that I've been wondering about.

    The distributive law holds for matrices. Let A and B be n x n matrices.
    Why is the following true for all A&B?

    ##(A+B)^2=A^2+2AB+B^2##

    I don't undrestand that middle term (2AB) and why there's a factor of 2 there, since matrices aren't always commutative (i.e., AB doesn't always equal BA).

    Shouldn't it be ##(A+B)^2=A^2+AB+BA+B^2## instead?

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Apr 11, 2015 #2

    robphy

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    As you say, AB doesn't always equal BA.
    So, as you say, ##(A+B)^2=A^2+AB+BA+B^2##

    But what you wrote isn't really the distributive law.
    ##C(A+B)=CA+CB##
     
  4. Apr 11, 2015 #3
    Yes, so it should be what you just stated right and not the first expression I had?
     
  5. Apr 11, 2015 #4

    Astronuc

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  6. Apr 11, 2015 #5
    It was from an example from a course I took. I did miss something however, the example had also stated that BA=0 in the problem statement....but that would still not yield the former equation. I do make mistakes sometimes when I copy down notes, so maybe I wrote the 2 in their by accident or my professor did....that would be my only explanation as the answer would be ##A^2+AB+B^2## if BA=0.
     
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