User Name Remember Me? Password

 Abstract Algebra Abstract Algebra Math Forum

 April 4th, 2010, 10:25 PM #1 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 matrix ring Consider the matrix ring M_2(Z_2) a.find the order of the ring, that is the number of elements in it. so Z_2= {0,1} this is a 2x2 matrix from M_2. So the orders of elements should be 4 which only has the elements 0 and 1. b.List all units in the ring, Since Z_2 has only 0,1 and 0 is not a unit because it is a 0 divisor of 2, therefore it leave 1 to be the unit. is this correct? April 5th, 2010, 09:42 AM   #2
Senior Member

Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: matrix ring

Quote:
 Originally Posted by tinynerdi Consider the matrix ring M_2(Z_2) a.find the order of the ring, that is the number of elements in it. so Z_2= {0,1} this is a 2x2 matrix from M_2. So the orders of elements should be 4 which only has the elements 0 and 1.
The order of the ring is the number of elements. There are 4 entries in each matrix, and 2 choices for each entry...

Quote:
 b.List all units in the ring, Since Z_2 has only 0,1 and 0 is not a unit because it is a 0 divisor of 2, therefore it leave 1 to be the unit.
The ring of interest in this case is M_2(Z_2), not Z_2. We want to know which elements are units... i.e., which have an inverse. So, we want invertible 2x2 matrices. What does an invertible matrix look like? April 5th, 2010, 11:45 AM #3 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: matrix ring a. it has the order of 8. b.invertible of 2x2 matrix is (a b) = A^-1 (c d) (d -b) x 1/(ad-bc) -c a) April 5th, 2010, 01:53 PM   #4
Senior Member

Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: matrix ring

Quote:
 Originally Posted by tinynerdi b.invertible of 2x2 matrix is (a b) = A^-1 (c d) (d -b) x 1/(ad-bc) -c a)
Right, but we want something a little easier to work with... Just thinking back to anything you know about linear algebra, how can you determine if a matrix is invertible? April 5th, 2010, 02:53 PM #5 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: matrix ring well to invertible then the det(M_2(Z_2) does not = 0. is that what you mean? Are we saying that the elements in the 2x2 matrix has any entries in the R? What does the Z_2 mean in this case if the entries in the M_2(Z_2) is not 0,1. April 5th, 2010, 04:15 PM   #6
Senior Member

Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: matrix ring

Right; a matrix is invertible iff it has non-zero determinant. You can prove that this holds for the nxn matrices over any field*. Z_2 is a field, and so the condition holds. There's no reason to worry about elements of R.
So... there are two elements of Z_2. i.e. we want det(M)=1 for invertibility.

*The theorem actually says that a matrix over a (commutative?) ring is invertible iff its determinant is invertible in the ring. Every non-zero element in a field...

Also, I missed this the first time:
Quote:
 it has the order of 8.
You have 2 choices for each. You have this 4 times. It should be 2*2*2*2 = 16 April 6th, 2010, 11:49 AM #7 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: matrix ring Therefore the unit is 1 because it has an inverse. I know this is not part of the question. But what if we work in Z_3 and Z_3 has {0,1,2}. So the det(M) can equal to 1 or 2, so the unit is 1 and 2. And if we work in Z_4, {0,1,2,3} then the unit is 1 and 3. 2 is not a unit because it is 2 = 0 mod 4. It is therefore 0 divisor. April 6th, 2010, 04:10 PM #8 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: matrix ring I'm not completely following what you're trying to say with the last post. Everything is correct, but I don't see the relevance. The only invertible element in Z_2 is 1, so a matrix A in M_2(Z_2) is invertible iff det(A)=1. Which elements of M_2(Z_2) have determinant 1? April 6th, 2010, 05:52 PM #9 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: matrix ring thanks.  Tags matrix, ring Search tags for this page
,

,

,

,

,

,

,

,

,

,

,

,

,

,

# prove that a in an invertible matrix in m(f) iff ad-bc does not equal zero

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post sebaflores Abstract Algebra 1 October 27th, 2013 04:29 PM tinynerdi Abstract Algebra 4 April 4th, 2010 10:17 PM xixi Abstract Algebra 0 January 30th, 2010 08:21 AM cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM Frazier001 Abstract Algebra 1 December 6th, 2007 01:21 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.      