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Prove Ab-ba=i Has No Solution

  1. Sep 7, 2006 #1
    Hi,I am a Chinese sophomore major in software engineering.I am reading Artin's Algebrarecently and have come across this problem in 1.1,and have been trying for 4 days in vain:cry:

    Give me some real thought guys,I will really appreciate it!
     
  2. jcsd
  3. Sep 7, 2006 #2
    Sorry,the problem should be this:prove AB-BA=I has no solution with realnumber where A and B are matrix
     
  4. Sep 7, 2006 #3

    AKG

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    I suppose you mean that you want to prove that there are no real square matrices A and B such that AB - BA = I. The key to proving this: Trace.
     
  5. Sep 8, 2006 #4
    It's probably way out of my area, but let's say you take a 1x1 matrix for A and a 1x1 matrix for B. Then A*B will equal B*A...because there are no additions or substractions etc inside the matrixes...and since they're both 1*1 they can be multiplied...wouldn't matrix I end up being just ?

    [tex][0][/tex]

    I probably said something very stupid...but it seems to me that it follows all the question parts...it's a matrix, it's real, it's a solution...and A and B can be anything...
     
  6. Sep 8, 2006 #5
    Real matrices are those matrices which have real entries. They are not necessarily 1x1.
     
  7. Sep 8, 2006 #6
    Oh so you have to prove the identity for all of them? I think what I was trying to do is find one case that works. Yeah...I'm way over my head...:(
     
  8. Sep 8, 2006 #7
    thanks!

    Yeah,using trace works!
    Another question:is trace invented just to prove this problem?I believe that a new concept usually come from a new mothod proving something.
     
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