Is the Trace Method the Key to Proving Ab-ba=i Has No Solution?

[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!

 

[GreenApple]

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

 

[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.

 

[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...

 

[Muzza]

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

 

[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...:(

 

[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.