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Homework Help: Invertible square matrix

  1. Feb 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the square matrix [tex]2N - I[/tex] is its own inverse if [tex]N^{2} = N[/tex]

    2. Relevant equations
    properties of invertible matrix

    3. The attempt at a solution
    I really don't know where to start here. I know that [tex](2N-I)(2N-I) = I[/tex], but where do I go on from there?
  2. jcsd
  3. Feb 28, 2010 #2
    You are correct in calculating (2N-I)(2N-I) and using N2=N in your calculations. If (2N-I)(2N-I)=I, then the desired result is achieved, as this implies [(2N-I)]-1=(2N-I).
  4. Feb 28, 2010 #3
    So the answer is simply [tex](2N-I)(2N-I) = I[/tex]? But how did I use [tex]N^{2} = N[/tex]?
  5. Feb 28, 2010 #4
    I just assumed you did :)
    The result (2N-I)(2N-I)=I is the desired result. So take (2N-I)(2N-I) and expand it. After you are done that, use N2=N.
  6. Feb 28, 2010 #5
    ok, so I get:

    [tex]2N^{2} - 2NI - 2NI + I^{2} = I[/tex]

    [tex]2N - 2N - 2N + I = I[/tex]

    But this doesn't work out.
  7. Feb 28, 2010 #6
    The coefficient on N2 is incorrect. It is not a 2.
  8. Feb 28, 2010 #7
    Ok I got it. Thanks for the help.
  9. Feb 28, 2010 #8
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