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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+B(I-AB)[tex]^{-1}[/tex] (I-AB)A
    Last edited: Dec 10, 2007
  2. jcsd
  3. Dec 10, 2007 #2


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    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
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