Register to reply

Disprove that AB-BA = I

by hamsterman
Tags: matrix, proof
Share this thread:
hamsterman
#1
Jan6-12, 05:21 AM
P: 74
The task is to prove that for no two matrices A and B, A*B - B*A = I, where I is the identity matrix.
I tried multiplying by the inverses of A or B, but that doesn't seem to lead to a more manageable form. The only way I see this could be done is by writing down all n*n (assuming n by n matrices) linear equations. It's easy to do when n = 2, but the same contradiction may not be as obvious for higher n.
I hope there is a more intelligent way to go about this.
Phys.Org News Partner Science news on Phys.org
Mysterious source of ozone-depleting chemical baffles NASA
Water leads to chemical that gunks up biofuels production
How lizards regenerate their tails: Researchers discover genetic 'recipe'
phyzguy
#2
Jan6-12, 06:11 AM
P: 2,179
What do you know about determinants?
hamsterman
#3
Jan6-12, 06:23 AM
P: 74
I know that det(AB) = det(BA), but I don't know what are the properties when subtraction is involved. Except for the case when only one line is different.

Borek
#4
Jan6-12, 06:46 AM
Admin
Borek's Avatar
P: 23,535
Disprove that AB-BA = I

Quote Quote by hamsterman View Post
I know that det(AB) = det(BA), but I don't know what are the properties when subtraction is involved.
Determinant is just a number, isn't it?
hamsterman
#5
Jan6-12, 07:05 AM
P: 74
What I mean is that I don't know what is det(AB-BA) even if I do know det(AB) and det(BA).
I'm looking at Sylvester's determinant theorem which looks related, but I still don't see a solution. Now I need to prove that for no M, det(M+I) = det(M), at least when M = AB.. (now that I think about it, there is probably no matrix that can't be written as a product of two others, is there?)
Dick
#6
Jan6-12, 08:54 AM
Sci Advisor
HW Helper
Thanks
P: 25,228
Quote Quote by hamsterman View Post
What I mean is that I don't know what is det(AB-BA) even if I do know det(AB) and det(BA).
I'm looking at Sylvester's determinant theorem which looks related, but I still don't see a solution. Now I need to prove that for no M, det(M+I) = det(M), at least when M = AB.. (now that I think about it, there is probably no matrix that can't be written as a product of two others, is there?)
Try taking the trace.
hamsterman
#7
Jan6-12, 09:30 AM
P: 74
So tr(AB-BA) = 0 ? Great. Thanks.


Register to reply

Related Discussions
Prove or disprove... Calculus 1
Subring: Prove or Disprove Calculus & Beyond Homework 1
Prove or Disprove: if a | bc, then a|b or a|c Calculus & Beyond Homework 13
Help Me Disprove This Biology 3
Prove or Disprove Linear & Abstract Algebra 10