Matrix

aqa · 2011-01-30 08:29:04

Row 1 (0, 1, *, 0, *, 0, *)
Row 2 (0, 0, 0, 1, *, 0, *)
Row 3 ( 0, 0, 0, 0, 0, 1, *)

Is a possible reduced echelon form for a 3 X 7 matrix. How many different such forms for a reduced row echelon matrix are possible if the the matrix is 2 X 3??

The above matrix is in reduced row echelon form.
Arbiitrary enteries are denoted by " * ".

I have no idea as to what is being asked or how to solve the problem.

Bob · 2011-01-30 11:18:59

hi aqa

Have a look at

http://en.wikipedia.org/wiki/Reduced_echelon_form

This may answer your query.

Bob

aqa · 2011-01-30 12:29:35

Thank you for the responce.

The following is an assumtion: If I only look at the first two rows and three columns. Then the answere is 0, because there is not a leading 1.

Is that how I am suppose to look at the problem??

Bob · 2011-01-30 20:33:59

hi aqa

I think it means start with an entirely new 2 x 3 matrix in rref. How many of these can you find? (From the form of the question you put * for anything that is not a zero or one and count all such as one possibility.)

Bob

Bob · 2011-01-31 02:03:02

hi aqa

It has got to be 40 years since I met reduced row echelon matrices (rrem) in my degree and I havent used them since, so Im a bit rusty.
Thats why (and it was late at night for me) I pointed you to Wiki. This site does get some criticism for lack of accuracy but Ive always found it good for mathematical topics.

Now Ive had a chance to think it through, Im ready to give you what is, I hope, a better answer.

The only use Ive come across for rrem is a way of solving simultaneous linear equations. (That is equations where there are no square, cube, or higher powers)

The procedure is no quicker or easier than other methods but it has the advantage that it is completely prescriptive ... you do exactly the same thing for every set of equations. So it is ideal if you want to write a computer program to solve equations. The Wiki page gives some pseudocode for this if you are interested and have the time to write and debug the program.

Ill demonstrate the procedure by way of an example. To use the procedure you must have all the equations in the form

Suppose the equations are

Step 1. Build a 3 by 4 matrix out of these numbers

Now to change into rrem

Step2. Divide the top row by its first element (2)

Step 3. Make new row 2 := 3 x row 1 - old row 2. Make new row 3 := row `1 - old row 3

later edit: Looking this over, I think this step may be back to front. The 'rust' had not quite cleared at this stage. But it works out ok so it doesn't matter a lot unless you are trying to build a computer procedure. If so, the later steps show the correct way to create the ones and zeros correctly.

This puts a one and two zeros into the first column.

Step 4.

Divide the second row by the second element in the row.

Step 5. Make new row 1 := old row 1 + 0.5 x row 2. Make new row 3 := old row 3 + 1.5 x row 2.

That has made the second column into a zero, a one and another zero.

Step 6. Divide row 3 by the third element in the row.

Step 7. Make new row 1 := old row 1 + 3/7 of row 3. Make new row 2 := old row 2 - 1/7 of row 3

Now you can translate that back into three equations

This procedure will always do that. If there are more equations than unknowns something funny will show whether the 'excess' equations are consistent with the ones you use to solve the equations. If you have insufficient equations then the procedure will not get completed.

Have you sorted out your original query? I have an answer if you are stuck.

Bob

Last edited by Bob (2011-01-31 02:09:53)

aqa · 2011-01-31 04:17:46

I am missing somthing, and it is probably right in front of me. I can't see the forest due to the trees.

Thank you for taking your time and thinking about the question/equation. I follow what you did with the reduced row echelon form. That makes sence. But I can not make a connection between the 3 x 7 matrix and a 2 x 3 matrix.
Can you tell me how you solved it???

Bob · 2011-01-31 04:34:30

hi aqa

Maybe I am wrong but I think there is a break between

Row 1 (0, 1, *, 0, *, 0, *)
Row 2 (0, 0, 0, 1, *, 0, *)
Row 3 ( 0, 0, 0, 0, 0, 1, *)
Is a possible reduced echelon form for a 3 X 7 matrix.

and the next bit

How many different such forms for a reduced row echelon matrix are possible if the the matrix is 2 X 3??

Firstly, you are told what ref is.

Secondly, you are asked about ref for another matrix entirely.

So now, just think about 2 by 3 matrices. Forget about the example.

So the general format for 2 by 3 is

but we want ref. so what could this be?

The *s could be anything. These are the only ones I can think of.

Bob

aqa · 2011-01-31 08:52:24

I think I am now seeing it. I had my mind wrapped around the origianl matrix.

Thank for your response. What do you think of these also.

(1, *, *) ( 0, 1, *) (1, *, 0) (0, 1, 0)
(0, 0, 0) (0, 0, 0) (0, 0, 1) (0, 0, 1)

Bob · 2011-02-01 01:42:51

hi aqa,

You seem to have been much more careful than me in considering all possible cases. I like all your suggestions.

Do you think I can still claim this as a total success for my teaching, as the 'pupil' has overtaken the 'teacher' ?

Bob

aqa · 2011-02-01 12:28:17

LOL, I still haven't figured out the wax on wax off thing yet. LOL

It wasn't until I saw what you were doing with the matrix that I finally understood the question. Thank You

Bob · 2011-02-01 21:31:18

hi aqa

You are very welcome,

Bob

Math Is Fun Forum

#1 2011-01-30 08:29:04

Matrix

#2 2011-01-30 11:18:59

Re: Matrix

#3 2011-01-30 12:29:35

Re: Matrix

#4 2011-01-30 20:33:59

Re: Matrix

#5 2011-01-31 02:03:02

Re: Matrix

#6 2011-01-31 04:17:46

Re: Matrix

#7 2011-01-31 04:34:30

Re: Matrix

#8 2011-01-31 08:52:24

Re: Matrix

#9 2011-02-01 01:42:51

Re: Matrix

#10 2011-02-01 12:28:17

Re: Matrix

#11 2011-02-01 21:31:18

Re: Matrix

Board footer