Question

Matrix question

Answer 1

sorry dont do matrix graphing :P

Answer 2

LOL

Answer 3

R u there?

Answer 4

HUH how did u get those equations?

Answer 5

What question r u reading?

Answer 6

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

Answer 7

recheck it it shldnt be abt traffic flow

Answer 8

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

Answer 9

Thanks lol

Answer 10

Sorry abt that. i thought I deleted b4 u came

Answer 11

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!

Answer 12

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

Answer 13

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 \[\left(\begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix}\right)^2=0\]

Answer 14

let me just solve that wait a sec

Answer 15

oh ya i see

Answer 16

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 \[\left(\begin{matrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{matrix}\right)^3=0\]

Answer 17

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

Answer 18

oh i see

Answer 19

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

Answer 20

Imperialist is on a roll :DDDD

Answer 21

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.

Answer 22

ok thanks for the explanation

Answer 23

I will page u if I have another question