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

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In summary, the conversation was about proving that there are no real square matrices A and B such that AB - BA = I. The key to proving this is using the concept of trace. The conversation also touched on the idea of real matrices and their properties.
  • #1
GreenApple
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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!
 
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  • #2
Sorry,the problem should be this:prove AB-BA=I has no solution with realnumber where A and B are matrix
 
  • #3
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
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...
 
  • #5
Real matrices are those matrices which have real entries. They are not necessarily 1x1.
 
  • #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...:(
 
  • #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.
 

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

What does "Prove Ab-ba=i Has No Solution" mean?

"Prove Ab-ba=i Has No Solution" is a mathematical statement that is asking for proof that the equation Ab-ba=i has no solution. This means that there cannot be any values for the variables A, B, and i that will make this equation true.

Why is it important to prove that Ab-ba=i has no solution?

Proving that Ab-ba=i has no solution is important because it helps us understand the limitations of the equation and the relationship between the variables. It also helps us avoid making false assumptions or conclusions based on this equation.

How do you prove that Ab-ba=i has no solution?

To prove that Ab-ba=i has no solution, we can use mathematical techniques such as algebraic manipulation, substitution, and contradiction. By manipulating the equation and trying different values for the variables, we can show that there is no solution that will make the equation true.

What are some possible reasons for Ab-ba=i having no solution?

One possible reason for Ab-ba=i having no solution is that the equation is inherently contradictory. For example, if the equation is A=2B and B=3A, it is impossible for both statements to be true at the same time, thus making the equation unsolvable. Another possible reason is that the variables are not defined in a way that allows for a solution. For instance, if the equation is A=2B and B=0, there is no value for A that will make both statements true.

Can a computer be used to prove that Ab-ba=i has no solution?

Yes, a computer can be used to prove that Ab-ba=i has no solution. Complex equations with multiple variables can be difficult for humans to solve, but computers can handle the calculations and manipulations more efficiently. However, it is important for a human to review and verify the results to ensure accuracy and avoid any errors in the programming or input of data.

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