Results 1 to 3 of 3

Math Help - Matrices went in 2x2

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    91

    Matrices went in 2x2

    Another challenge problem, inspired by an earlier question.

    Given arbitrary 2x2 matrices A and B, prove that

    AB+BA=\beta A+\alpha B+(\gamma-\alpha\beta)I

    where \alpha is the trace of A, \beta is the trace of B and \gamma is the trace of AB or of BA. I is of course the 2x2 identity matrix.

    Although a brute force approach would work, a more elegant solution would be welcome.

    Enjoy.

    Moderator approved. CB
    Last edited by CaptainBlack; August 22nd 2010 at 09:19 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    We use the Cayley-Hamilton identity for A, \ B, \ A+B

    A^2-\alpha A+\det A\cdot I_2=O_2

    B^2-\beta B+\det B\cdot I_2=O_2

    (A+B)^2-Trace(A+B)+\det (A+B)\cdot I_2=O_2

    We have Trace(A+B)=Trace(A)+Trace(B)=\alpha+\beta

    and \det(A+B)=\det A+\det B+\gamma-\alpha\beta

    Then

    A^2+B^2+AB+BA-\alpha A-\alpha B-\beta A-\beta B+\det A\cdot I_2+\det B\cdot I_2+(\gamma-\alpha\beta)I_2=O_2

    or AB+BA=\beta A+\alpha B+(\gamma-\alpha\beta)I_2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    91
    Quote Originally Posted by red_dog View Post
    (A+B)^2-Trace(A+B)+\det (A+B)\cdot I_2=O_2
    Should be

    (A+B)^2-Trace(A+B)\cdot(A+B)+\det (A+B)\cdot I_2=O_2

    but otherwise spot on! Thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 25th 2010, 07:34 PM
  2. Total matrices and Commutative matrices in GL(r,Zn)
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: August 16th 2010, 03:11 AM
  3. Matrices help!!
    Posted in the Algebra Forum
    Replies: 0
    Last Post: October 30th 2009, 03:21 AM
  4. Matrices Help
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 24th 2009, 10:36 PM
  5. Matrices represented by Symmetric/Skew Symmetric Matrices
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: October 25th 2008, 06:06 PM

Search Tags


/mathhelpforum @mathhelpforum