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Linear Homogeneous Systems

  1. Oct 2, 2009 #1
    1. The problem statement, all variables and given/known data


    If a linear homogeneous system Ax=0 has a non - trivial solution and A is an n x n matrix, then (choose ALL correct answers)

    A. A has rank less than n
    B. Each system Ax=b with the same coefficient matrix A has a solution
    C. A is row equivalent to I
    D. If Ax=b has one solution it has many
    E. A is invertible

    If a linear homogeneous system Ax = 0 has a non - trivial solution and A is n x n, then (choose ALL correct answers)

    A. A is invertible
    B. A has rank less than n
    C. A is row equivalent to I
    D. If Ax = b has one solution it has many
    E. Each system Ax = b with the same coefficient matrix A has a solution

    3. The attempt at a solution

    So I looked through my notes for these two questions, found some properties and chose the answers the notes led me to believe were true, but I got both incorrect. I only have one attempt left on each question so I want to make sure I'm 100% sure on the answers Ibefore I try. Could anyone give me a nudge in the right direction? ie. explain linear homogeneous systems?
     
  2. jcsd
  3. Oct 2, 2009 #2
    what were the properties that led you to the incorrect answers? What were those answers?

    It would help if I knew what I needed to explain.
     
  4. Oct 2, 2009 #3
    My notes said that a square matrix is also invertible, and is row equivalent to I
     
  5. Oct 2, 2009 #4
    Oh, ok. Only a square matrix can be invertible. Not all of them are. Your notes must be mistaken.
     
  6. Oct 2, 2009 #5
    So even though it's a square it isn't necessarily invertible?
     
  7. Oct 2, 2009 #6
    precisely. Read https://www.physicsforums.com/showpost.php?p=2369970&postcount=22" post. I wrote it for somebody that was really being dense so if it seems like it's talking down to you a little, don't take it personally ;)

    It gives the simplest examples possible of a matrix equation where the matrix is not invertible. If you have any questions, I'll be happy to answer.
     
    Last edited by a moderator: Apr 24, 2017
  8. Oct 2, 2009 #7
    Okay thanks for the link. But I'm still kind of lost about the other choices
     
  9. Oct 2, 2009 #8
    The only true statement in both cases is that A has rank less than n.
     
  10. Oct 2, 2009 #9
    There's one more true statement. Which is of course equivalent to the one that you mentioned.
     
  11. Oct 3, 2009 #10
    Sorry what? So A has rank less than n, and there is one more true statement other than that?
     
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