# Homework Help: Simple Linear Algebra Proof

1. Feb 25, 2010

### maherelharake

1. The problem statement, all variables and given/known data
Prove that if A and B are symmetric nxn matrices then AB is symmetric if and only if AB=BA

2. Relevant equations

3. The attempt at a solution
I tried to say AB=(AB)T=BTAT=BA
But I don't think this is correct.

2. Feb 25, 2010

### vela

Staff Emeritus
When you have an "if and only if," you need to prove both directions. You have to show:

1) If AB is symmetric, then AB=BA.
2) If AB=BA, then AB is symmetric.

What you have so far is a proof of #1. You assumed AB is symmetric, which means AB=(AB)T, and found AB=BA. So you're half done. You just need to prove #2 now.

3. Feb 25, 2010

### maherelharake

Would the second part be...
AB=BA=(BA)T=ATBT=AB

4. Feb 25, 2010

### mmmboh

If A=AT and B=BT, and if AB is symmetric,
then $$(AB)^T=AB=A^TB^T=(BA)^T$$
thus for AB to be symmetric, (AB)T=(BA)T therefore AB=BA.
Almost the same thing you wrote.

For the second part, if AB=BA, then (AB)T=....

Last edited: Feb 25, 2010
5. Feb 25, 2010

### vela

Staff Emeritus
No, because you don't know that BA=(BA)T.

6. Feb 25, 2010

### maherelharake

Why can AB=(AB)T but we don't know if BA=(BA)T

7. Feb 25, 2010

### mmmboh

Just because $$(AB)^T=AB$$, doesn't mean $$(BA)^T=BA$$

8. Feb 25, 2010

### vela

Staff Emeritus
You don't know AB=(AB)T either. That's what you're trying to prove. When you're proving #2, all you know is A and B are symmetric and AB=BA.

9. Feb 25, 2010

### maherelharake

AB=BA=BTAT=(AB)T

Would this work?

10. Feb 25, 2010

### vela

Staff Emeritus
Can you justify each step?

11. Feb 25, 2010

### maherelharake

Well we assume AB=BA. We can say that the next step holds too since we know that A and B are symmetric. The final step is just a property. Is that flawed?

12. Feb 25, 2010

### vela

Staff Emeritus
Nope, that's perfect.

13. Feb 25, 2010

### maherelharake

Great thanks again. Whenever you get a chance, I entered another post over on the previous thread. If you can help me verify, it will be great! (even though you have already helped tremendously)