Results are out! Find what you need...fast. Get quick advice or join the chat
Hey there Sign in to join this conversationNew here? Join for free

Question on Groups concerning matrices...

Announcements Posted on
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    Let M denote the set of real 2x2 matrices of the form \begin{pmatrix} x & y \\ -y & x \end{pmatrix} where x and y are not both zero,

    M= \{\begin{pmatrix} x & y \\ -y & x \end{pmatrix}: x^2 + y^2 \not= 0\}.
    Show that M is a group under matrix multiplication. (you may assume that matrix multiplication is associative)

    I am just confused about something, I am trying to figure if for example this is multiplication of the form:

    \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\begin{pmatrix} c & d \\ -d & c \end{pmatrix}

    or if the operation has to be of the form

    \begin{pmatrix} a & b \\ -b & a \end{pmatrix}\begin{pmatrix} a & b \\ -b & a \end{pmatrix},


    ---------------------------

    nevermind, i messed up on something really simple... :rolleyes:
    • 4 followers
    Offline

    It's safe to assume that G is a group since the question asks you to show that it is, it doesn't ask whether or not it is.

    If your example of \begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix} didn't work then you must have multiplied your matrices wrong.

    The closure property is this: if A, B \in G then AB \in G.

    So suppose A = \begin{pmatrix} x & y \\ -y & x \end{pmatrix} and B = \begin{pmatrix} s & t \\ -t & s \end{pmatrix}, with x^2+y^2 \ne 0 and s^2+t^2 \ne 0. Show that AB can be written in the form \begin{pmatrix} p & q \\ -q & p \end{pmatrix} and that p^2+q^2 \ne 0 (where p,q are in terms of x,y,s,t).

    P.S.: To fix your LaTeX code, take the { symbols off the start of each of the lines.
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    (Original post by nuodai)
    It's safe to assume that G is a group since the question asks you to show that it is, it doesn't ask whether or not it is.

    If your example of \begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 2 \\ -2 & 3 \end{pmatrix} didn't work then you must have multiplied your matrices wrong.

    The closure property is this: if A, B \in G then AB \in G.

    So suppose A = \begin{pmatrix} x & y \\ -y & x \end{pmatrix} and B = \begin{pmatrix} s & t \\ -t & s \end{pmatrix}, with x^2+y^2 \ne 0 and s^2+t^2 \ne 0. Show that AB can be written in the form \begin{pmatrix} p & q \\ -q & p \end{pmatrix} and that p^2+q^2 \ne 0 (where p,q are in terms of x,y,s,t).

    P.S.: To fix your LaTeX code, take the { symbols off the start of each of the lines.
    Ah thanks, coding fixed now
    I actually worked out 2x3 + (1x-2) as being 9 for some reason in my counterexample which is why it confused me :s but my counterexample actually does conform to the group rules

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: April 7, 2012
New on TSR

Have a UCAS application question?

Post it in our dedicated forum

Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.