# Showing a subset does not form a subgroup (explaination wanted)

• Oct 10th 2009, 08:57 AM
elninio
Showing a subset does not form a subgroup (explaination wanted)
If not sure how to show that a subset does not form a subgroup. Here is the example:

Let $\displaystyle G=GL_2(R)$. Show that the subset S of G defined by
$\displaystyle S={[ a b c d ]|b=c}$
of symmetric 2x2 matrices deos not form a subgroup of G.

I'm not sure where to start. I understand the three conditions:
i)ab is an element of S
ii) e is an element of S
iii)a^-1 is an element of S
But i'm not used to physically working with matrices to prove these. How do I show that at least one of these is not satisfied for S?
• Oct 10th 2009, 09:59 AM
HallsofIvy
Quote:

Originally Posted by elninio
If not sure how to show that a subset does not form a subgroup. Here is the example:

Let $\displaystyle G=GL_2(R)$.

You can't use HTML tags inside LaTex. Use [ math ]G= GL_2(\bold{R})[ /math ] (without the spaces) to get $\displaystyle G= GL_2(\bold{R})$.

Quote:

Show that the subset S of G defined by
$\displaystyle S={[ a b c d ]|b=c}$
use "[ math ]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[ math ]" to get "$\displaystyle \begin{bmatrix}a & b \\ c & d\end{bmatrix}$"

Quote:

of symmetric 2x2 matrices deos not form a subgroup of G.

I'm not sure where to start. I understand the three conditions:
i)ab is an element of S
ii) e is an element of S
iii)a^-1 is an element of S
But i'm not used to physically working with matrices to prove these. How do I show that at least one of these is not satisfied for S?
As for (ii), the identity is $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ wjhich is a symmetric matrix.

As for (iii), the inverse of $\displaystyle \begin{bmatrix} a & b \\ b & c\end{bmatrix}$. a "general" symmetric matrix, can be shown to be $\displaystyle \frac{1}{ac- b^2}\begin{bmatrix}c & -b \\ -b & a\end{bmatrix}$, also a symmetric matrix.

So I recommend you focus on (i). Is the product of two symmetric matrices symmetric? What is $\displaystyle \begin{bmatrix} a & b \\ b & c\end{bmatrix}\begin{bmatrix}d & e \\ e & f\end{bmatrix}$?
• Oct 10th 2009, 06:46 PM
elninio
Pardon me, I was away all day.

$\displaystyle \begin{bmatrix}da+eb & ea+fb \\ db+ec & eb+fc\end{bmatrix}$

Can I say that this does not satisfy i) since the product of two symmetric matrices is not a symmetric matrix?
Does the question imply that my symmetric matrices should be abbd and accd? Since b=c, wouldnt this imply a symetric product matrix? Or am I completely on the wrong page on this one...
• Oct 11th 2009, 07:03 AM
HallsofIvy
Quote:

Originally Posted by elninio
Pardon me, I was away all day.

$\displaystyle \begin{bmatrix}da+eb & ea+fb \\ db+ec & eb+fc\end{bmatrix}$
Yes, that's the whole point. Since, in general, $\displaystyle ea+fb\ne db+ec$, the set of symmetric matrices is not closed under multiplication.
More specifically, both $\displaystyle \begin{bmatrix}2 & 3 \\ 3 & 1\end{bmatrix}$ and $\displaystyle \begin{bmatrix}3 & 2 \\ 2 & 1\end{bmatrix}$ are symmetric matrices and their product is $\displaystyle \begin{bmatrix}2 & 3 \\ 3 & 1\end{bmatrix}\begin{bmatrix}3 & 2 \\ 2 & 1\end{bmatrix}= \begin{bmatrix}12 & 7 \\ 11 & 7\end{bmatrix}$ which is NOT a symmetric matrix.