Prove Ab-ba=i Has No Solution

1. Sep 7, 2006

GreenApple

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

Give me some real thought guys,I will really appreciate it!

2. Sep 7, 2006

GreenApple

Sorry,the problem should be this:prove AB-BA=I has no solution with realnumber where A and B are matrix

3. Sep 7, 2006

AKG

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.

4. Sep 8, 2006

Robokapp

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 ?

$$[0]$$

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...

5. Sep 8, 2006

Muzza

Real matrices are those matrices which have real entries. They are not necessarily 1x1.

6. Sep 8, 2006

Robokapp

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...:(

7. Sep 8, 2006

GreenApple

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.