Invertible Matrices

Let's suppose I have a matrix BA which is the ordered product of the matrices B and A.

If I claim that BA is invertible, then do I need that B and A are invertible too?

Also, if I have that the inverse of A, A^-1 exists, then can I say the right inverse and left inverse of A also exist?
I mean I know there are plenty of left/right inverses of A, but A^-1 is one of them right, so if it exists, surely I can say A has a left and right inverse?

Think about the determinant of BA and facts you know about the determinant.

|BA| = |B| |A|, if any of |A| or |B| are 0, then |BA| = 0. So, in order for |BA| to be invertible, A and B must both be non-invertible and their inverses must exist. Thank you!!!

What about the right/left inverses part?

Let's suppose I have a matrix BA which is the ordered product of the matrices B and A.

If I claim that BA is invertible, then do I need that B and A are invertible too?

Also, if I have that the inverse of A, A^-1 exists, then can I say the right inverse and left inverse of A also exist?
I mean I know there are plenty of left/right inverses of A, but A^-1 is one of them right, so if it exists, surely I can say A has a left and right inverse?

A is invertible if there's B s.t. AB = BA = I.

Are A and B necessarily square? You can only define determinants for square matrices.

Hi, yes for the first part, A and B are necessarily square. For the second part, no, because left and right inverses also exist for non-square matrices.

Hi, yes that is the definition.

Your whole question is solved by reinstating the definitions.

I suggest you pay attention to the definitions.

Not really. For BA to be invertible means there is C such that CBA = I = BAC.

From this it’s clear A has a left inverse and B has a right inverse but you need some result to know that means A and B are invertible.

Hi, just to clarify. A may not have an inverse yet it may have a left/right inverse - that I know. Here A has a left inverse, what is the result that means A also has an inverse too? I mean is it just because we are assuming A and B are square matrices, so if it has a left inverse, it also has an inverse = left inverse = right inverse.

Let's suppose I have a matrix BA which is the ordered product of the matrices B and A.

If I claim that BA is invertible, then do I need that B and A are invertible too?

Also, if I have that the inverse of A, A^-1 exists, then can I say the right inverse and left inverse of A also exist?
I mean I know there are plenty of left/right inverses of A, but A^-1 is one of them right, so if it exists, surely I can say A has a left and right inverse?

(1) Assuming A, B are square, as someone mentioned: det(BA)=det(B)det(A). Invertible <--> non-zero determinant for square matrices. Hence, if BA is invertible, then det(B) and det(A) must be non-zero - that is, A, B must be invertible too.

(2) Property of square matrices: saying that A has an inverse, or a left inverse, or a right inverse are equivalent. Consider matrix A with inverse B. Then, from definition that BA=AB=I, we see that B is left and right inverse of A. You can also prove the reverse statement i.e. if left and right inverse exists, then inverse exists: let left inverse be B and right inverse C. Then, B=BI=B(AC). By associativity, B(AC)=(BA)C=IC=C.