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Matrix A s.t. A^3=0 but A^2 doesn't

  1. Sep 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Find a matrix, A, such that A3=0, but A2≠0.


    2. Relevant equations



    3. The attempt at a solution
    I don't actually think this is possible. My only other thought is to take A2*A where this product equals zero, and then decompose A. But I don't know how to do that such that it is the "square root" of A.
     
  2. jcsd
  3. Sep 30, 2012 #2
    Try a 3x3 matrix with 1s above the diagonal.
     
  4. Sep 30, 2012 #3
    How would I have known to look for that?

    From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?
     
  5. Sep 30, 2012 #4
    Let A be an n xn matrix for which A^k = 0 for some k > n. Prove that A^n = 0.

    This theorem tells me to work on nxn matrices since I know I know I can construct one that is A^n = 0. So your case we needed A^3. Hence I picked a 3x3
     
  6. Sep 30, 2012 #5

    Ray Vickson

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    Homework Helper

    Re "How would I have known to look for that?" I'll bet your book or notes have some relevant material. Alternatively, you could Google 'nilpotent matrix', but I guess you would need to know that is what such A are called.

    RGV
     
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