## anonymous 4 years ago Matrix question

1. anonymous

2. anonymous

sorry dont do matrix graphing :P

3. anonymous

LOL

4. anonymous

R u there?

5. anonymous

HUH how did u get those equations?

6. anonymous

7. anonymous

I accidentally posted the wrong question but then I deleted it an reposted the right one

8. anonymous

recheck it it shldnt be abt traffic flow

9. anonymous

Ah, I clicked too fast before you did that, my bad. I'll look at your actual question now!

10. anonymous

Thanks lol

11. anonymous

Sorry abt that. i thought I deleted b4 u came

12. anonymous

Ah yes, a very cool problem! Let me think of what answer to give you. What class are you in actually, since that might decide how much depth I should go into!

13. anonymous

I am in linear algebra but I am only in second chapter/second week

14. anonymous

Okay, I'll just spit out a bunch of things and we'll go from there. First off, 2x2 cases are boring, so I'll probably go a bit higher for some things I say! Note that $\left(\begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix}\right)^2=0$

15. anonymous

let me just solve that wait a sec

16. anonymous

oh ya i see

17. anonymous

But note that you also have that $\left(\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}\right)^2=0$ and you have that $\left(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}\right)^3=0$

18. anonymous

In fact, any matrix that is strictly upper triangular (meaning that the main diagonal and everything below it is zero) is nilpotent!

19. anonymous

oh i see

20. anonymous

However, not all nilpotent matrices are of this form. Note that $\left(\begin{matrix} 12 & -18 \\ 8 & -12 \\ \end{matrix}\right)^2=0$

21. anonymous

Imperialist is on a roll :DDDD

22. anonymous

I'm sure you will learn a lot more about this matrices later in your class, I will tell you two things you should notice about all of them. 1. All of them have determinant = 0 2. If A is the nilpotent matrix and k is the earliest integer such that A^k=0, then the trace of A, A^2, A^3, ..., A^(k-1)=0. Since trace(A^m) for m greater than or equal to k is obviously zero (since all of those matrices are zero), then trace(A^m)=0 for all m>0.

23. anonymous

ok thanks for the explanation

24. anonymous

I will page u if I have another question