Thread: Conflicting defintions/explanations

My professor states in one instance that if is invertible, then when , leading to . Yet later on in class, during a proof involving the multiplicity of invertible matrices, he makes the claim that implies that is invertible when , leading to . I understand this theorem fairly easily, but my professor made a big deal about whether the order of the equation is or , yet he clearly contradicts himself if order is so important. A similar is in once instance, leads to , and then later on states that it is that leads to . Again, I understand fairly easily that is a matrix is an inverse of another matrix, then the inverse of that first matrix is the other matrix, but this professor is pedantic.

So I need clarification on the theorem. If then is invertible and , or does imply is invertible and therefore . My professor is a real hardass about following the proofs EXACTLY as they are stated, but like I said, the two claims that he made seem to contradict themselves if order is so important.

Originally Posted by Pinkk

My professor states in one instance that if is invertible, then when , leading to . Yet later on in class, during a proof involving the multiplicity of invertible matrices, he makes the claim that implies that is invertible when , leading to . I understand this theorem fairly easily, but my professor made a big deal about whether the order of the equation is or , yet he clearly contradicts himself if order is so important. A similar is in once instance, leads to , and then later on states that it is that leads to . Again, I understand fairly easily that is a matrix is an inverse of another matrix, then the inverse of that first matrix is the other matrix, but this professor is pedantic.

So I need clarification on the theorem. If then is invertible and , or does imply is invertible and therefore . My professor is a real hardass about following the proofs EXACTLY as they are stated, but like I said, the two claims that he made seem to contradict themselves if order is so important.

if is a square matrix, say then is invertible if and only if or for some matrix the important point, which unfortunately is not mentioned in most elementary

linear algebra courses, is that for square matrices we have if and only if anyway, in this case we call the inverse of and similarly would be the inverse of

so for square matrices the following four statements are equivalent, meaning any of them implies the other three:

1.

2.

3.

4.

for non-square matrices things are different. an matrix could have left inverse, i.e. for some matrix but no right inverse, i.e. for any

matrix it is also possible that has a right inverse but not a left inverse (or of course might have neither left nor right inverse). but if has both left and right inverse, then it has to be

a square matrix, i.e. and then the left and right inverse must be equal.

I understand that implies given the necessary conditions, but my professor used a proof to show this where he stated the definition I mentioned in my first post. But he is not consistent with that definition in the proof. The definition he first gave was implies is invertible and thus . He uses this defintion to show that , but as I mention in my first post, he somewhat "contradicts" himself because he put great emphasis on the order of those equations.

Originally Posted by Pinkk

I understand that implies given the necessary conditions, but my professor used a proof to show this where he stated the definition I mentioned in my first post. But he is not consistent with that definition in the proof. The definition he first gave was implies is invertible and thus . He uses this defintion to show that , but as I mention in my first post, he somewhat "contradicts" himself because he put great emphasis on the order of those equations.

in my previous post i actually ignored what your prof said because it's nonsense. i also said that the definition is equivalent to saying that or because for square matrices

if and only if anyway, let's just stick to your prof's definition: then in order to show that we need to show that which is obviously the same as

so, yes, the order is not important and you're right. another point that i mentioned is the "uniqueness", i.e. if and then now we can

talk about "the" inverse.

Ah okay, thank you. And yes, I had a hunch that my professor was making it more convoluted than it has to be. The if and only if statement applies only to square matrices since performing would result in an and results in an , and therefore , and if are square matrices of equal size , then since for square matrices.

see how an MIT prof explains the inverse of a matrix. he mentions what i said, i.e. for square matrices if and only if but he also says that the proof is not "easy",

(well, it's not terribly hard either! haha) and so he skips it. i like this guy! his lecture is very accurate, although he talks too much! haha watch the video from 21':30" if you like:

MIT OpenCourseWare | Mathematics | 18.06 Linear Algebra, Spring 2005 | Video Lectures | detail

No doubt someone can prove me wrong, but I don't think there is such a thing as a one-sided inverse for square matrices. If AB=I then BA=I.
But in a more abstract setting there are certainly monoids in which one can have AB=I but not BA=I. But this can only happen in rather restricted circumstances:
it can't happen for a finite monoid
if AB=I but BA != I then there is no C such that BC=I nor D such that DA=I. If there were then it is easy to show that AB=BA=I.

Originally Posted by alunw

No doubt someone can prove me wrong, but I don't think there is such a thing as a one-sided inverse for square matrices. If AB=I then BA=I.
But in a more abstract setting there are certainly monoids in which one can have AB=I but not BA=I. But this can only happen in rather restricted circumstances:
it can't happen for a finite monoid
if AB=I but BA != I then there is no C such that BC=I nor D such that DA=I. If there were then it is easy to show that AB=BA=I.

For instance, the Bicyclic Monoid: .