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Homework Help: Algebraic Properties of Matrix Operations

  1. Jan 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A and B be (2x2) matrices such that A^2 = AB and A does not equal the zero matrix O. Find the flaw in the following proof that A = B:

    Since A^2 = AB, A^2 - AB = the zero matrix O
    Factoring yields A(A-B) = O
    Since A does not equal O, it follows that A - B = O.
    Therefore, A = B.



    3. The attempt at a solution

    I tried setting up two matrices A and B where A = [ a b, c d] and B = [ e f, g h] and following through on the steps of the proof to see if each of the statements was true. However, I kept finding that they were all true.

    Please help.
    Thanks.
     
  2. jcsd
  3. Jan 19, 2010 #2

    Mark44

    Staff: Mentor

    Problem is, the conclusion is not true for all matrices A and B, even when neither is the zero matrix.

    Try playing with matrices that have mostly (but not all) zero entries.
     
  4. Jan 20, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    This is not true. The fact that a product of matrices is 0 does NOT imply one of the factors must be 0.
    For example
    [tex]\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}= \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}[/tex]
    the 0 matrix.

     
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