Results 1 to 5 of 5

Math Help - Linear Algebra Counterexample

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    91

    Linear Algebra Counterexample

    If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

    I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hello
    Quote Originally Posted by ah-bee View Post
    If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

    I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.
    AB+BA=0 \Longleftrightarrow AB=-BA

    Multiplying both sides by BAB yields  ABBAB=-BABAB which you can transform into A^2B^3=B^3A^2 using AB=-BA.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ah-bee View Post
    If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

    I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

    Thanks.
    Left multiply bothe sides by A and right multiply both sides by B^2:

    AB = -BA \Rightarrow A^2 B^3 = -A(BA)B^2

    Substitute -AB = BA into the right hand side:

    A^2 B^3 = -(AB)AB^2 \Rightarrow A^2 B^3 = BA A B^2.

    Keep playing that game:

    A^2 B^3 = BA (AB) B \Rightarrow A^2 B^3 = BA (-BA) B = -B(AB)(AB) = -B (-BA) (-BA)

    = -B^2 (AB) A = -B^2 (-BA) A = B^3 A^2.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Smile

    Quote Originally Posted by ah-bee View Post
    If AB + BA = 0 then A^2B^3 = B^3A^2 if it is true give a short proof if it is true give a counterexample

    I'm sure that the implication stated above is wrong but im struggling to give a counter-example. Anybody care to give me a hand in forming a counter-example or would you like to point out that the statement is indeed true and give me a proof of it.

    Thanks.
    What makes you so sure that it is false?

    Remember that matrix multiplication is associative (for matrices over the reals):

    A^2B^3 = AABBB = A(AB)BB

    =A(-BA)BB = -ABABB = -(AB)(AB)B = -BABAB

    = -B(AB)(AB) = -BBABA = -BB(AB)A

    =BBBAA = B^3A^2
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2008
    Posts
    91
    darn... thats a lot of multiplication. never wouldve seen that tbh. thanks for all the help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 1st 2011, 11:00 PM
  2. Linear Algebra: Linear Independence question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 3rd 2011, 06:28 AM
  3. Replies: 2
    Last Post: December 6th 2010, 04:03 PM
  4. Replies: 7
    Last Post: August 30th 2009, 11:03 AM
  5. Replies: 3
    Last Post: June 2nd 2007, 11:08 AM

Search Tags


/mathhelpforum @mathhelpforum