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A small matrix proof

  1. Jan 16, 2007 #1

    radou

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    Let A and B be square matrices of order n, such that B = 2A - I. (1) One has to proove that B is involutory (B^2 = I) <=> A is idempotent (A^2 = A).

    Starting off with direction <= , one multiplies (1) with A from the right and from the left, separately, and obtains the results BA = A, and AB = A, respectively. This implies B = I, and I is an involutory matrix, so that direction is prooved. (I hope, that's why I'm asking.)

    The second direction, => , causes trouble and I'd appreciate any help. I tried multiplying (1) with B, but it didn't seem to help. Thanks in advance.
     
  2. jcsd
  3. Jan 16, 2007 #2

    StatusX

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    BA = A and AB = A does not give B = I. For example, take A=0 (there are other examples too). It is true that B=I if BA = A and AB = A holds for all A, but that isn't the case here.

    Just square both sides of B=2A-I. This should give both directions pretty easily.
     
  4. Jan 16, 2007 #3

    radou

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    Just realized that, thanks.

    Did that, and prooved both directions. Thanks again! :smile:
     
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