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Question on Groups concerning matrices...

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    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:
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    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.
  3. 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

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