- #1
Bipolarity
- 776
- 2
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
- 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