Linear algebra square matrices

[std10093]

Homework Statement

A and B are two matrices n X n

Homework Equations

AB-BA=identity matrix
It is critical for me to prove that the are no matrices that are capable to hold the above equation true

The Attempt at a Solution

I made several efforts.I had the idea to get the main diagonal of AB and the main diagonal of BA same and so AB-BA would gine a main diagonal foul of zeros.But this idea was not good enough...

 

[Dick]

Do you know what the 'trace' of a matrix is? Take the trace of both sides.

 

[std10093]

I found the solution!
trace(AB-BA)=trace(I)
trace is linear so trace(AB) - trace(BA) = trace(I)
Easily we prove that trace(AB)=trace(BA) so we get the following equation
0= n * 1 ,n is the dimension of matrices n x n
This of course is incongruous.

 

[std10093]

Do you know what the 'trace' of a matrix is? Take the trace of both sides.

Now i saw what you posted.I did not knew.It's my first semester at university and my first lessons of linear algebra.Professor gave that exercise only to me in order to find it myself.I did'n know what trace is ,so what i named as main diagonal ,finally is the trace(this is what i was told by the professor when i gave to him the solution)

Anyway thanks a lot