# Does AB=I imply BA=I?

by asmani
Tags: imply
 P: 84 Hi all. Does there exist a field F such that AB=I doesn't imply BA=I, where A and B are square matrices (both n×n) over the field F? Thanks in advance.
 PF Patron HW Helper Sci Advisor Thanks P: 25,490 hi asmani! hint: BAB ?
 HW Helper Sci Advisor P: 2,020 AB=I always implies BA=I regardless of the field. This is because the rank-nullity theorem holds over any field, so an injective linear map (between vector spaces of the same dimension n) is necessarily surjective (and vice-verse). In fact AB=I will imply BA=I even over a commutative ring. Probably the quickest way to see this is to use the characterization "An nxn matrix with entries in a commutative ring R is invertible (i.e. has a left and right inverse) iff detA is a unit in R". Note that AB=I implies that detA and detB are units. P.S. Here I'm addressing the problem of showing that "left invertible" implies "right invertible". Once you have this, you can use tiny-tim's hint to show that the left inverse is equal to the right inverse.
P: 47

## Does AB=I imply BA=I?

AB=I does not always imply BA=I . There are somethings called left inverses and right inverses. You can look it up when you see the least square projections of a vector which does not lie on a column space S onto S.

Then if there is a left inverse C , then, though , CA = I but AC ≠ I .
A similar story goes for the right inverse.

Good Luck
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 Quote by vish_maths AB=I does not always imply BA=I . There are somethings called left inverses and right inverses. You can look it up when you see the least square projections of a vector which does not lie on a column space S onto S. Then if there is a left inverse C , then, though , CA = I but AC ≠ I . A similar story goes for the right inverse. Good Luck
We are talking about square matrices. In that case, AB=I does certainly imply BA=I.
P: 47
 Quote by micromass We are talking about square matrices. In that case, AB=I does certainly imply BA=I.
oops :) yep . Sorry , i was talking in general terms :) thanks for quoting !
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 Quote by micromass We are talking about square matrices. In that case, AB=I does certainly imply BA=I.
Not always. There are certain mathematical characteristics that are required to be able to say that. I would have agreed completely had you instead said "We are talking about square matrices over a field."

 Quote by tiny-tim hint: BAB ?
This is a very nice hint. What mathematical characteristics let tiny-tim get away with writing BAB without using any parentheses? What characteristics do you get "for free" just by saying the word "field"?
P: 84
Thanks a lot for the replies.
 Quote by morphism Here I'm addressing the problem of showing that "left invertible" implies "right invertible". Once you have this, you can use tiny-tim's hint to show that the left inverse is equal to the right inverse.
Actually the original problem was to show that "left invertible" implies "right invertible" and vice versa.
 Quote by tiny-tim hint: BAB ?
 Quote by D H This is a very nice hint. What mathematical characteristics let tiny-tim get away with writing BAB without using any parentheses? What characteristics do you get "for free" just by saying the word "field"?
I can't see how to work out this yet. I know that we can write B=BI and since AB=I, we get to B=B(AB) which is the same as B=(BA)B (commutativity of multiplication) and then we have B(I-BA)=0. Does it help?
P: 1,167
 Quote by asmani Thanks a lot for the replies. Actually the original problem was to show that "left invertible" implies "right invertible" and vice versa. I can't see how to work out this yet. I know that we can write B=BI and since AB=I, we get to B=B(AB) which is the same as B=(BA)B (commutativity of multiplication) and then we have B(I-BA)=0. Does it help?
Careful! If you had commutativity _matrix-wise_ (you do have it entry/element-wise),
AB=I would automatically imply BA=I.
 HW Helper Sci Advisor P: 9,371 this is not a trivial result. it is equivalent to showing that any set of n independent vectors in k^n must span it.
P: 84
 Quote by Bacle2 Careful! If you had commutativity _matrix-wise_ (you do have it entry/element-wise), AB=I would automatically imply BA=I.
Sorry, I meant associativity, which leads to B(AB)=(BA)B.
Thanks.
 P: 84 Here is the proof I found: AB=I implies det(A)det(B)=1, which means that det(A)≠0. It's a known fact that A is invertible iff det(A)≠0. Thus A is invertible and has both right and left inverses and they are equal. Is there something wrong?
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