1. Not finding help here? Sign up for a free 30min 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!

Invertible Matrix Theorem

  1. May 7, 2008 #1
    General question regarding the Inv. Matrix Thm:

    One part of the theorem states that for an nxn invertible matrix, then there exists at least one solution for each b in Ax=b. Why wouldn't it be "there exists at MOST one solution for each b" since every column/row has a pivot. How would there exist more than one solution for each b if the columns span R_n?
     
  2. jcsd
  3. May 7, 2008 #2

    Defennder

    User Avatar
    Homework Helper

    Are you sure that's what the theorem says? My book doesn't say "at least one". It says "exactly one". Here's what Wikipedia says:

     
  4. May 7, 2008 #3
    Oh, weird. Yeah my book does say "at least one solution". Thanks for showing me the wiki entry though.
     
  5. May 8, 2008 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Well, the statement is still true: if a matrix is invertible, then the equation Ax= b has exactly one solution so it is certainly true that there is at least one solution.

    If A is not invertible then Ax= b may have no solutions or an infinite number of solutions.

    You book may have some reason for emphasizing "the solution exists" right now rather than "the solution is unique"- both of which are true for A invertible.
     
  6. May 17, 2010 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    First, don't "hijack" someone else's thread for your own question- that's rude. Use the "new topic" button to start your own thread.

    Second, go back and reread the question. You can't prove any of those, they are all false. For example, if A is the 0 matrix, "Ax= b" has NO soution for b non-zero and has an infinite number of solutions if b is 0.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Invertible Matrix Theorem
  1. Invertible Matrix (Replies: 1)

  2. Invertible matrix (Replies: 0)

  3. Invertible matrix (Replies: 3)

  4. Invertible matrix (Replies: 11)

Loading...