Results 1 to 2 of 2

Math Help - Invertible Matrices

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    9

    Invertible Matrices

    Hello!

    This is one problem I've been tackling for a while:

    If A, B, and A+B are all invertible matrices of order "n", prove that A^{-1} + B^{-1} is invertible and that:

    (A^{-1}+B^{-1})^{-1} = A(A+B)^{-1}B = B(A+B)^{-1}A

    First, is there a generic formula for A^{-1} + B^{-1}? (If so, our instructor never taught us one). I'm also assuming that we can't assume AB is commutable, because that could make everything so much easier. ^_^;

    Anyway. Any helpful hints might help. Basically what I've tried to do so far is equate the "middle" and RHS statements by multiplying both sides by inverses to isolate (A+B)^{-1}, but I don't really see how that 'proves' anything ...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by ginarific View Post
    Hello!

    This is one problem I've been tackling for a while:

    If A, B, and A+B are all invertible matrices of order "n", prove that A^{-1} + B^{-1} is invertible and that:

    (A^{-1}+B^{-1})^{-1} = A(A+B)^{-1}B = B(A+B)^{-1}A

    First, is there a generic formula for A^{-1} + B^{-1}? (If so, our instructor never taught us one). I'm also assuming that we can't assume AB is commutable, because that could make everything so much easier. ^_^;

    Anyway. Any helpful hints might help. Basically what I've tried to do so far is equate the "middle" and RHS statements by multiplying both sides by inverses to isolate (A+B)^{-1}, but I don't really see how that 'proves' anything ...

    How do you show a matrix is invertible? Find a matrix that multiplied by the first one gives you the identity one!
    Well, let us show that A^{-1}+B^{-1} is the inverse of the matrix A(A+B)^{-1}B:

    A(A+B)^{-1}B(A^{-1}+B^{-1})=A\left[(A+B)^{-1}B+(A+B)^{-1}A\right]A^{-1}= A(A+B)^{-1}(B+A)A=I and we're done!

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. invertible matrices
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 5th 2010, 11:48 PM
  2. Invertible matrices
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 9th 2009, 05:57 PM
  3. Invertible Matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 15th 2009, 11:19 PM
  4. invertible matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 15th 2009, 05:28 AM
  5. invertible matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 21st 2005, 01:32 PM

Search Tags


/mathhelpforum @mathhelpforum