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Nullspace of a square matrix A and A^2 are related?

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Say that A is a square matrix. Show that the following statements are true, or give a counter example:
    a) If x is in the nullspace of A, then x is in the nullspace of A2
    b) If x is in the nullspace of A2, the x is in the nullspace of A.

    2. Relevant equations

    3. The attempt at a solution

    I solved part a, or maybe I didn't. I said

    "Ax=0 is our assumption.
    A2x = A*Ax = A(0) = 0
    so statement a is true."

    However, for part b, I stated:

    "A2x=0 is our assumption.
    Let B=A2, so Bx=0 is true.
    A*Ax = 0

    We have no way of knowing if Ax is true yet.
    However if we left multiply by the inverse of A,
    we can see that Ax=0. Therefore the statement
    b is true unless the determinant of A is zero,
    and the inverse does not exist."

    However, when trying any and all matrices, some with and some without a determinant equal to zero, and finding the nullspace of the matrix squared and checking it with the original matrix, it always returns a matrix of zero. Ideas? Thanks in advance.
  2. jcsd
  3. Mar 12, 2013 #2


    Staff: Mentor

    Part a looks fine.
    Part b - see if you can find a matrix A (2 x 2 is fine) such that A2 = 0, even though A ≠ 0.
  4. Mar 12, 2013 #3
    Oh... Okay I get it now.

    If I use the matrix

    0 0
    0 1 = A and

    0 0
    0 0 = A2

    The nullspace of A2 has infinite solutions
    and the nullspace of A will have at least one x value
    that will have to be zero in order for it to be a valid
    equation, so the A2 nullspace can not
    transfer over.

    Thanks :)
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