1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
  2. jcsd
  3. Jan 19, 2010 #2


    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


    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.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook