Proofs on Idempotent Matrices

Date: 05/09/2000 at 02:07:07
From: Laura Jones
Subject: Abstract Algebra: Idempotents

Hello. My final exam in abstract algebra is tomorrow and this problem 
appeared on an exam early in the semester, but I'm a little confused 
about it. Can you explain and help me work through it?

An element e of a ring is said to be IDEMPOTENT if e^2 = e.

  (a) Determine all idempotent 2x2 matrices over R

  (b) Prove that in an integral domain, the only idempotents are 
      0 and 1.

I think I've got part (b):

         e^2 = e
     e^2 - e = 0
      e(e-1) = 0
so
           e = 0   or   
           e = 1

This is what I've done for part (a):

     [ a b ] [a b ]  =  [ a b ]
     [ c d ] [c d ]     [ c d ]
so
     [ a^2 + bc   ab + bd  ]  =  [ a b ]
     [ ac + ad    bd + d^2 ]     [ c d ]
so 
     a^2 + bc  = a

     ac  + ad  = c
        a(c+d) = c

     ab  + bd  = b
        b(a+d) = b
           a+d = 1
             a = 1-d

     bd  + d^2 = d

then

     (1-d)(1-d) + bc = (1-d)

But now what?

            [1 0]     [0 0]
I know that [0 1] and [0 0] are idempotents, but there are more. What 
are they?

Thanks for any help you may be able to offer. I'm sure the exam won't 
have anything to do with this problem but I would still like to 
understand it!

Laura

Date: 05/09/2000 at 13:53:35
From: Doctor Rob
Subject: Re: Abstract Algebra: Idempotents

Thanks for writing to Ask Dr. Math, Laura.

You have started on a path which can get you the right answer, but 
you've made a couple of errors.

     [ a b ] [ a b ] = [ a b ]
     [ c d ] [ c d ]   [ c d ]
so
     [ a^2 + b*c   a*b + b*d ]  =  [ a b ]
     [ a*c + c*d   b*c + d^2 ]     [ c d ]

(notice two changes from your work above), so 

     a^2 + b*c = a

     a*c + c*d = c
       c*(a+d) = c
             c = 0   or   a + d = 1

     a*b + b*d = b
       b*(a+d) = b
             b = 0   or   a + d = 1

     b*c + d^2 = d

If a + d = 1 is false, then you can see that b = 0 and c = 0, so 
a^2 = a and d^2 = d. The solutions here are the two you gave above, 
namely I and 0.

If a + d = 1 is true, then you get the equations

       d = 1 - a

     b*c = a - a^2
         = a*(1-a)
         = a*d
         = (1-d)*d
         = d - d^2.

Pick any value for a you want, make d = 1 - a, and then pick b and c 
so that b*c = a*d. That will give you an idempotent matrix. 
Furthermore, every idempotent matrix will either be I, 0, or one 
formed in this way.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/