Question

How would I find the rank and nullity of the following matrix? (Given in comments)

Answer 1

\[\left[\begin{matrix}2 & 2 & 0 & 2 \\ 2&3&0&3 \\ 1&2&0&3 \\ 1& 0 &0&0\end{matrix}\right]\]

Answer 2

Am I correct in thinking that to find the nullity, I first find the dimension of ker(A) [assuming the above matrix is 'A'], by taking the augmented matrix, and finding all possible answers to Ax=0?

Answer 3

Then rank(A) + nullity(A) = m for an n x m matrix?

Answer 4

that is correct.

Answer 5

Would you possibly be able to help with augmenting the matrix etc? I'm getting a kind of strange answer :/

Answer 6

sure. We need to row reduce that matrix first. Im not pro enough to type the matrices on here >.< so i'll do it on paper and post. one sec.

Answer 7

Thanks so much :)

Answer 8

The matrix needs to be in row reduced echelon form (i think thats how you spell it.)

Answer 9

I'm getting an answer where the bottom row is [0 0 0 -6] ... Don't think thats correct :/

Answer 10

Sorry, -2, not -6 ... Not helpful, regardless...

Answer 11

alright, im double checking this on my phone, but this is what im getting for the RREF of A.

Answer 12

Yeah thats right. do you have any questions about anything in those three pictures?

Answer 13

Just one; In the second row of your final answer, how did [0 1 0 1] become [0 1 0 0]?

Answer 14

i used the 1 in row 3 to kill it. I did:
\[-R_3+R_2\]

Answer 15

Alrighty! So the non-leading variable is only "x_4" (if the notation is similar to what you're familiar with) ... This is where I get really stuck - the Ax=0 thing :/

Answer 16

As far as your question is concerned, all the information we need is in that RREF form.

The number of pivots is the rank of A, and the number of free variables is the Nullity of A.

Answer 17

Oh really?? So I dont have to let the non-leading variable be denoted some constant ('t' or something) and find the possible solutions to Ax=0?? :D

Answer 18

you could still get the answer by doing that, but what you are interested in is the dimension of the Null Space, not really the Null Space itself.

If the question was, "Find a basis for the Null Space", or "Describe the Ker(A)", then i would do that.

In this problem, just knowing its one-dimensional (since there is only one freee variable) is enough.

Answer 19

Great! Thanks so much!

Answer 20

Morning Zarkon!

Answer 21

morning

Answer 22

nice work

Answer 23

thanks :) oh, i found out a few days ago there might be a Complex Variables class in the spring (pending enough people register). i remember you talking about that class, i hope i get to take it lol >.<

Answer 24

nice...definitely take it if you can. very Interesting stuff

Answer 25

Haha Zarkon thanks for your help on almost exactly the same question earlier today :P Much obliged :)

Answer 26

np

Answer 27

well I hope you all enjoy your day...as for me I get to be in meetings all day :(