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!(adsbygoogle = window.adsbygoogle || []).push({});

- 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

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# Some questions about invertibility of matrix products

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