# Homework Help: Rank of a matrix and more

1. Sep 6, 2007

### rock.freak667

1. The problem statement, all variables and given/known data
Find the rank of the matrix A,where
$$A= \left( \begin{array}{cccc} 1 & 1 & 2 & 3\\ 4 & 3 & 5 & 16\\ 6 & 6 & 13 & 13\\ 14 & 12 & 23 & 45 \end{array} \right)$$

Find vectors$$x_0$$and$$e$$ such that any solution of the equation

$$Ax= \left( \begin{array}{c} 0\\ 2\\ -1\\ 3 \end{array} \right)$$ $$(*)$$
can be expressed in the form $$x_0+\lambdae$$ where $$\lambda\epsilonR$$

Hence show that there is no vector which satisfies $$*$$ and has all its elements positive

2. Relevant equations

First attempt at such a question, so unknown are any relevant equations

3. The attempt at a solution
Well for the first part to get the rank I put A in RRE form and then counted the number of non-zero rows and got for so $$r(A)=4$$

now for the second part,I thought to solve the equation by multiplying by $$A^{-1}$$ and finding $$x$$ but then I realized that I have no idea where to get $$x_0$$ or $$\lambda$$ or $$e$$

can anyone show me how to do these types of questions or can show me some similar example?

2. Sep 6, 2007

### Hurkyl

Staff Emeritus
Well, you made a mistake somewhere in here.

You might have guessed that -- if you can write any solution in the form the problem asks for, what does the rank of the matrix have to be?

(Hint: what does the nullity of the matrix have to be?)

3. Sep 6, 2007

### rock.freak667

Did I do the row-reduction wrong?
well from wikipedia...$$rank(A)+Nullity(A)=n$$ well $$n=4$$ in this case

BTW...This is the first time I have heard of nullity

4. Sep 6, 2007

### Hurkyl

Staff Emeritus
I believe so. The statement of the problem implies the rank is not 4. (In fact, it implies a specific number for the rank) I tried once to do the row reduction myself, and I got the number I expected.

5. Sep 6, 2007

### rock.freak667

Well I believe I did it over correctly and got $$r(A)=3$$

6. Sep 6, 2007

### rootX

yes, you seems to be correct, if this is what you were trying to get:
$$\pmatrix{1 & 1 & 2 & 3\cr 0 & 1 & 3 & -4\cr 0 & 0 & 1 & -5\cr 0 & 0 & 0 & 0}$$

use maxima!!

http://aycu21.webshots.com/image/27020/2000682090404007350_rs.jpg

Last edited: Sep 6, 2007
7. Sep 6, 2007

### rock.freak667

But how do I use the fact that $$r(A)=3$$ and the nullity to find the vectors in that form?

8. Sep 6, 2007

### Hurkyl

Staff Emeritus
Well, how do you normally solve systems of equations? Have you tried that?

9. Sep 6, 2007

### rock.freak667

Well normally for that matrix I would just augment it and try to put it in RRE form but then i dont know where $$x_0$$ and $$e$$ and $$\lambda$$ comes in

10. Sep 6, 2007

### Hurkyl

Staff Emeritus
Well, try solving it first, then think about it.

By the way, you can edit your original post to fix that one formula; you're supposed to put spaces between things. And it looks a lot nicer if you use [ itex ] instead of [ tex ] for stuff in paragraphs.