Results 1 to 8 of 8

Math Help - [SOLVED] some simple inverse matrix help

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    28

    [SOLVED] some simple inverse matrix help

    1. IF matrices A, B and C are such that A = B^{-1} C , Show that the inverse of  C^{-1}B is A

    2. In this question i was given 2 matrices A and B . it tells me to work out AB which i know how to do, and then it says hence determine B.
    how do i find the inverse of B from the product AB, im thinking i have to use I somewhere, do i make AB = I ?

    i would appreciate if you can contribute your thinking process aswell. Im pretty good at all the matix calculations but when i see non standard questions like these which requres thinking, i just don't know where to start from and it really kills my confidence

    Thanks!
    Last edited by llkkjj24; March 6th 2010 at 05:11 AM. Reason: solved
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2009
    From
    Alberta
    Posts
    173
    Quote Originally Posted by llkkjj24 View Post
    1. IF matrices A, B and C are such that A = {B^-1}  C , Show that the inverse of  C^-1 B is A
    A handy property of inverses (and transposes!) is the following:

     (AB)^{-1} = B^{-1}A^{-1} , note the reverse order!

    So if we have;

    A = B^{-1}C and we want to show that the inverse of C^{-1}B is A, We just take the inverse of it.

    (C^{-1}B)^{-1} = B^{-1}[(C^{-1})^{-1}] = B^{-1}C = A

    Does this help?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,583
    Thanks
    1418
    Quote Originally Posted by llkkjj24 View Post
    1. IF matrices A, B and C are such that A = {B^{-1}}  C , Show that the inverse of  C^{-1} B is A
    What is A(C^{-1}B)? What is (C^{-1}B)A?
    (To get all of -1 as superscript put it in braces: {-1}.)
    2. In this question i was given 2 matrices A and B . it tells me to work out AB which i know how to do, and then it says hence determine B.
    how do i find the inverse of B from the product AB, im thinking i have to use I somewhere, do i make AB = I ?
    What did you get for AB?

    i would appreciate if you can contribute your thinking process aswell. Im pretty good at all the matix calculations but when i see non standard questions like these which requres thinking, i just don't know where to start from and it really kills my confidence

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    May 2009
    Posts
    28
    Quote Originally Posted by Kasper View Post
    A handy property of inverses (and transposes!) is the following:

     (AB)^{-1} = B^{-1}A^{-1} , note the reverse order!

    So if we have;

    A = B^{-1}C and we want to show that the inverse of C^{-1}B is A, We just take the inverse of it.

    (C^{-1}B)^{-1} = B^{-1}[(C^{-1})^{-1}] = B^{-1}C = A

    Does this help?
    thanks, just wondering, is there a way to prove this using identities?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    May 2009
    Posts
    28
    Quote Originally Posted by HallsofIvy View Post
    What is A(C^{-1}B)? What is (C^{-1}B)A?
    (To get all of -1 as superscript put it in braces: {-1}.)

    What did you get for AB?
    wow, working it out makes a huge difference, so i mutiplied A and B, all 3x3 matrices, and got back an identity matrix.

    that means matrix A is the inverse of B right?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,583
    Thanks
    1418
    Where did you get that these are 3 by 3 matrices? You are not just looking at specific examples are you? And, no, I see nothing here to imply that A is the inverse of B.

    You are told that A= B^{-1}C and asked to show that A is the inverse of C^{-1}B. Two matrices, X and Y, are inverse to each other if and only if XY= YX= I, the identity matrix. To show that "A is the inverse of C^{-1}B", you need to show that A(C^{-1}B)= I and that [tex](C^{-1}B)A= I[tex].

    Of course, A(C^{-1}B)= B^{-1}C(C^{-1}B) and (C^{-1}B)A= (C^{-1}B)B^{-1}C. Use the "associative law" to manipulate those.

    For the second problem, you said you were able to find AB but you still have not said what you got.
    Last edited by HallsofIvy; March 4th 2010 at 11:51 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    May 2009
    Posts
    28
    thanks, for no 2 i was given 2 matrices A and B


    I was told to work out AB and hence determince B^{-1}

    I mulitplied A and B and got a matrix back. The identity matrix


    so since A . B = I , then A must be B^{-1}?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,583
    Thanks
    1418
    Well, you should also show that BA= I but for square matrices if AB= I then BA= I also.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Finding the inverse of a matrix using it's elementary matrix?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 7th 2011, 06:08 PM
  2. [SOLVED] Derivative of a matrix inverse and matrix determinant
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 24th 2011, 08:18 AM
  3. [SOLVED] Matrix Inverse
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: April 12th 2010, 12:17 PM
  4. [SOLVED] Inverse matrix with adj(A)
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 10th 2010, 06:42 PM
  5. [SOLVED] Finding inverse matrix
    Posted in the Advanced Algebra Forum
    Replies: 11
    Last Post: January 17th 2007, 10:41 AM

Search Tags


/mathhelpforum @mathhelpforum