What is the Ring Generated by R-Linear Combinations of These Matrices?

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In summary: R.In summary, the set of four matrices 1 0 | 0 1 | -1 0 | 0 1 0 1 | 1 0 | 0 1 | -1 0 are closed under multiplication and generate the ring of 2x2 matrices over the real numbers. They are not closed under addition and are not a ring themselves.
  • #1
plxmny
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This set of four matrices:

1 0 | 0 1 | -1 0 | 0 1
0 1 | 1 0 | 0 1 | -1 0

are closed under multiplication. What is it called? I know that it is not
those silly quaternions
 
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  • #2
plxmny said:
This set of four matrices:

1 0 | 0 1 | -1 0 | 0 1
0 1 | 1 0 | 0 1 | -1 0

are closed under multiplication. What is it called? I know that it is not
those silly quaternions

Is it the Klein 4-group?
http://en.wikipedia.org/wiki/Klein_four-group
 
  • #3
Thanks, NateTG, for taking the time to respond. That is what I thought until I realized that it had to be a RING for closure. So then I thought I would like to know the NAME of it, at least.
 
  • #4
If it's a ring, where's zero, and what are the addition and multiplication operations?
 
  • #5
NateTG,

0 is just

0 0
0 0

and + is just matrix addition and x is just matrix multiplication.

I found some stupid wikipedia thing where these 4 were given names like
K0 K1 K2 K3 but no references were given so that was a dead end.

I'm at work so I can't look it up in my books.
 
  • #6
Your set is not closed under addition, and thus not a ring.
 
  • #7
repeat after me ... "I don't know" ... all together now ... "I don't know"...
 
  • #8
So, you mean the ring *generated* by those four elements?
 
  • #9
plxmny said:
repeat after me ... "I don't know" ... all together now ... "I don't know"...

I don't know... what you're trying to say, or what your question is if you have one.
 
  • #10
... but name is not so important ... as long as you don't intend to lookup an index ...
 
  • #11
he means multiplication group of course instead of ring
 
  • #12
The last one squared isn't a member of the group. if it was
0 -1
-1 0

then you'd have the klein 4 group (well, isomorphic to it)
 
  • #13
1) it isn't a ring
2) it isn't closed under multiplication.
3) assuming you really mean 'what is the ring generated by R-linear combinations of these elements' (and that you're operating over R), those elements generate all od M_2(R) the 2x2 matrix ring
 

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