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If A^2 = 0, then A is not an invertible matrix

  1. Feb 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix

    2. Relevant equations


    3. The attempt at a solution
    We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not invertible, since ##0## has no inverse. This is a contraction, so it must be the case that if ##A^2 = 0##, then ##A## is not invertible.

    Is this the way I should be doing this problem?
     
  2. jcsd
  3. Feb 14, 2017 #2

    fresh_42

    Staff: Mentor

    I first thought - and this might well have been intended - that you should show, that there is a non-trivial element in the kernel of ##A##, namely the entire image of ##A##, but I like your solution better.
     
  4. Feb 15, 2017 #3

    Math_QED

    User Avatar
    Homework Helper

    Looks good to me as well. Note that you can prove this directly by using determinants, but I suspect you are not allowed to use determinants at this stage.
     
  5. Feb 15, 2017 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    There are nonzero matrices so as A2=0. You should prove that they are not invertible.
    For example, the square of the following matrix is zero.
    \begin{pmatrix}

    0 & 1 \\
    0 & 0

    \end{pmatrix}
     
    Last edited: Feb 15, 2017
  6. Feb 16, 2017 #5
    If A^2 = 0 and A is invertible, this implies A^(-1) A^2 = A^(-1) 0 = 0. No need to bother with non-invertible A's here.
     
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