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
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
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: 23,363
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: 25,251
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