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Question on Groups concerning matrices...

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:

(edited 12 years ago)

Reply 1

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.