Question

What is the rank of a matrix?

Answer 1

THe dimension of column space A

Answer 2

In simple terms it's asking you how many pivot points are in the matrix.

Answer 3

When you try to find the Column A you reduce to echelon form the matrix and find the pivot columns and those pivot columns make up Col A.

Answer 4

Rank of a matrix is an integer \(n\ge 1 \) such that there is at least one minor of order n which is not singular and every minor greater than \( n \) is singular.

Answer 5

Okay, how would you use this to determine whether a linear system with the given augmented matrix has a unique solution infinitly many solutions or no solutions?

Answer 6

Well if you have a square matrix if the rank is equal to n then you have a unique solution.

Answer 7

By using Rouché–Capelli theorem.

http://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem

Answer 8

Never heard of that theorem weird.

Answer 9

In simple words. You have a matrix A mXn .
The matrix is unique if the Rank A=n
The matrix is infinity if the Rank A<n
For no solution you need more information.

Answer 10

Can you give me more examples!

Answer 11

M=
3 -2 0 1 | 1
1 2 -3 1 |-1
2 4 -6 2 | 0

Answer 12

You are trying to find the Rank M?

Answer 13

Or see if it has a unique or infinite solutions?

Answer 14

Yes

Answer 15

to see whether the linear system with the given augmented matrix has a unique solution infinitly many solutions or no solutions

Answer 16

Well you don't really need to know rank in order to know the solution. Do you know how to reduce it to echelon form?

Answer 17

yeah, but i need to know how to do it using the rank method so i know how to use it in general...

Answer 18

Well the answer is no solution but you can't know that with simply using rank.

Answer 19

okay, i see, thanks

Answer 20

http://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem
see you can know if there is a unique solution or infinite
but you can't know if there is no solution.

Answer 21

if your teacher told you that you can know if there is no solution by knowing the rank of the matrix I suggest you challenge him/her =)

Answer 22

lol, will do!

Answer 23

Make sure you have your best troll face on when you do.

Answer 24

Actually you can know, if a system is inconsistent (i.e., no solution exists) via rank of the coefficient matrices.

Answer 25

Ref: http://www.math.tamu.edu/~fnarc/psfiles/rank2005.pdf

Answer 26

I guess you can but like I said before I see no advantage in using Rank to know if an augmented matrix has a solution or not. Seems like a waste of work.

Answer 27

Like for your example I can tell that there is no solution because the second row is a scalar multiple of the third and if you row reduce that you get a zero row for the matrix that equals a value which means no solution. Again I don't know the Rank and doing so will require more work,

Answer 28

you said something else before and this is a very useful tool.

Answer 29

@SockPuppet WOW! something just clicked in my head!!!