Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A small matrix proof

  1. Jan 16, 2007 #1


    User Avatar
    Homework Helper

    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


    User Avatar
    Homework Helper

    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


    User Avatar
    Homework Helper

    Just realized that, thanks.

    Did that, and prooved both directions. Thanks again! :smile:
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: A small matrix proof