Some questions about invertibility of matrix products

[Bipolarity]

After solving some problems about matrix invertibility and learning some theorems (and proving them), I have developed a set of questions about matrix invertibility. I have some claims but I don't know if they're true or false, so I was wondering if someone could point out which ones are true and which ones are false. Please don't give me any counterexamples or proofs, I wish to do them myself!

- A and B are not necessarily square, unless explictly stated
- The products AB and BA are defined wherever they happen to be mentioned

Here's what I already know:
- If A and B are invertible, the product AB and the product BA are both invertible, if they are defined.

What about the following?

1) If A and B are singular matrices, is the product AB also singular?
2) If A is invertible, but B is singular, is AB invertible or singular? What about BA?
3) If AB is invertible, can we conclude anything about the invertibility of A and/or B?
4) If AB is singular, can we conclude anything about the invertibility of A and/or B?
5) If BA is invertible, can we conclude anything about the invertibility of A and/or B?
6) If BA is singular, can we conclude anything about the invertibility of A and/or B?
7) If we know that A and B are square matrices, how does that affect Question 3?
8) If we know that A and B are square matrices, how does that affect Question 4?
9) If we know that A and B are square matrices, how does that affect Question 5?
10) If we know that A and B are square matrices, how does that affect Question 6?

Again, I only want to know whether they are true or false. I would like to prove/find counterexamples myself.

BiP

 

[Erland]

After solving some problems about matrix invertibility and learning some theorems (and proving them), I have developed a set of questions about matrix invertibility. I have some claims but I don't know if they're true or false, so I was wondering if someone could point out which ones are true and which ones are false. Please don't give me any counterexamples or proofs, I wish to do them myself!

- A and B are not necessarily square, unless explictly stated
- The products AB and BA are defined wherever they happen to be mentioned

Here's what I already know:
- If A and B are invertible, the product AB and the product BA are both invertible, if they are defined.

What about the following?

1) If A and B are singular matrices, is the product AB also singular?
2) If A is invertible, but B is singular, is AB invertible or singular? What about BA?
3) If AB is invertible, can we conclude anything about the invertibility of A and/or B?
4) If AB is singular, can we conclude anything about the invertibility of A and/or B?
5) If BA is invertible, can we conclude anything about the invertibility of A and/or B?
6) If BA is singular, can we conclude anything about the invertibility of A and/or B?
7) If we know that A and B are square matrices, how does that affect Question 3?
8) If we know that A and B are square matrices, how does that affect Question 4?
9) If we know that A and B are square matrices, how does that affect Question 5?
10) If we know that A and B are square matrices, how does that affect Question 6?

Again, I only want to know whether they are true or false. I would like to prove/find counterexamples myself.

BiP

The notions of invertibility and singularity for matrices are only defined for square matrices. Therefore, it has no meaning to say that a nonsquare matrix is invertible / noninvertible / singular / nonsingular. Therefore, the questions 1-6 have no meaning if the matrices are not square.

For a square matrix, invertible is the same thing as nonsingular, and singular is the same thing as noninvertible.

For the questions 7-10, the answers are:

7) Both A and B are invertible/nonsingular.
8) At least one of A and B is singular/noninvertible.
9) Same as 7).
10) Same as 8).

See http://en.wikipedia.org/wiki/Invertible_matrix

 

[Bipolarity]

The notions of invertibility and singularity for matrices are only defined for square matrices. Therefore, it has no meaning to say that a nonsquare matrix is invertible / noninvertible / singular / nonsingular. Therefore, the questions 1-6 have no meaning if the matrices are not square.

For a square matrix, invertible is the same thing as nonsingular, and singular is the same thing as noninvertible.

For the questions 7-10, the answers are:

7) Both A and B are invertible/nonsingular.
8) At least one of A and B is singular/noninvertible.
9) Same as 7).
10) Same as 8).

See http://en.wikipedia.org/wiki/Invertible_matrix

Thank you so much Erland! A grave mistake on my part not to realize that only square matrices have inverses (both left and right inverses). I will not attempt to prove the 4 results.

BiP