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!

Non singular

  1. Dec 10, 2007 #1
    Prove that if the matrix I-AB is non-singular, then so i I-BA.
    This was one of my test questions and got 3 points off. Can anyone tell me what I did wrong.

    my proof:
    assume I-AB is nonsingular
    then (I-AB)[tex]^{-1}[/tex] exists

    Let C=(I-AB)[tex]^{-1}[/tex]
    Consider (I+BCA)(I-BA)=I-BA+BCA-BCABA
    =I-BA+BC(I-AB)A
    =I-BA+B(I-AB)[tex]^{-1}[/tex] (I-AB)A
    =I-BA+BIA
    =I-BA+BA
    =I
     
    Last edited: Dec 10, 2007
  2. jcsd
  3. Dec 10, 2007 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    3 points over how many?

    Maybe you lost some points because

    i) You did not prove that if C is the left inverse of A, then it is also a right inverse. So you might have lost some points because you only showed that (I+BCA) is the left inverse of (I-BA).

    ii) The definition of nonsingular in your book is that A is nonsingular if its determinant is nonzero. And so, you might have lost some points because you did not explain that since I-BA had an inverse, then its determinant was nonzero and hence I-BA is nonsingular.

    But other than that, your proof looks very good! go see your instructor.
     
    Last edited: Dec 10, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Non singular
  1. Singular Matrices (Replies: 12)

  2. Degree of singularity (Replies: 11)

Loading...