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Matrix algebra proof

  1. Mar 9, 2015 #1
    This problem is so simple that I'm not exactly sure what they want you to do:

    Let A and B be n x n matrices such that AB = BA. Show that (A + B)^2 = A^2 + 2AB + B^2. Conclude that (I + A)^2 = I + 2A + A^2.

    We don't need to list properties or anything, just manipulate. This all seems self-evident from the distributive property, and showing that I^2 = I.
  2. jcsd
  3. Mar 9, 2015 #2


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    Science Advisor
    Gold Member

    If AB = BA
    and (A+B)^2 = (A+B)(A+B)
    then the rest more or less falls into place.
  4. Mar 9, 2015 #3
    So AB = BA
    (A+B)(A+B) = A^2 + 2AB + B^2
    II = I
    (I+A)(I+A) = I^2 + 2AI + A^2 = I + 2A + A^2

    Would this probably be what they're looking for? Not sure how much more in detail I can go
  5. Mar 10, 2015 #4


    Staff: Mentor

    I think you need some more detail here. Is it important that AB = BA in your proof?
    I think you need some more detail here as well, particularly in how you expand (I + A)(I + A).
  6. Mar 10, 2015 #5


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    Science Advisor
    Gold Member

    First of all, great name. Then, like Mark said, just expand the product term-by-term, without grouping.
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