# A very basic question about matrix operations

1. May 20, 2014

### Regtic

This is a very basic question that I've been too embarassed to ask my elementary linear algebra teacher this far into the course:

Lets say we have a matrix

$$\begin{pmatrix} 1 & 2 \\ 2 & 1 \\ \end{pmatrix}$$

Why can't we do something like $R_2 - 2R_1$ and $R_1 - \frac {1}{2} R_2$ at the same time? We would be multiplying each row by a multiple of the other. My teacher always does more than one row operation at a time. What is it that I'm missing?

Last edited: May 20, 2014
2. May 20, 2014

### SammyS

Staff Emeritus

3. May 20, 2014

### Regtic

I edited it and forgot to fix the the plurality. Sorry.

4. May 20, 2014

### SammyS

Staff Emeritus
Don't you mean that you teacher never does more than one row operation at a time ?

5. May 20, 2014

### Regtic

No he does it all the time, are you not allowed? I've done it on tests and never got marks off for it.

6. May 20, 2014

### Regtic

I guess that explains that. LOL

7. May 20, 2014

### Regtic

Are we 100% sure it's not allowed even if you follow some rules? I know he does it, there must be some rules he's following so he doesn't mess up

8. May 20, 2014

### SammyS

Staff Emeritus
Well, if your teacher does more than one row operation at a time, I would think it is clear that it's fine to do.

And, yes it is fine to do. Just be careful.

9. May 20, 2014

### Regtic

Careful how? Is the only thing we need to watch out for accidently creating a row/column of zeros?

10. May 20, 2014

### SammyS

Staff Emeritus
Yes, that sort of thing.

11. May 20, 2014

### Ray Vickson

You CAN do more than one row operation at the same time, provided that you do not overwrite the old entries by the new ones---doing that would create the unsolvable dilemma of modifying row 2 by adding a multiple of row 1 but modifying row 1 by adding a multiple of row 2 (but row 2 has already been modified using row 1 and row 1 ha been modified by row 2 .... it just never ends). The two operations you pose would be done at the same time by left-multiplying by the matrix
$$\pmatrix{1 & -1/2 \\-2 & 1}$$

12. May 20, 2014

### Regtic

So I just can't modify a row that I'm adding to another one?

13. May 21, 2014

### Ray Vickson

It would depend on the order in which you do things, and whether or not you overwrite old entries with new ones, etc.

14. May 21, 2014

### Regtic

That sounds vague. There's no way to know if I'm messing up without experience?

Edit: I think I kind of get it though, I've been doing it for a while. I was just wondering why it wouldn't work if I did both of those operations at the same time. What I'm really doing is multiple matrix operations in one matrix, and I just need to keep track of which coefficients I'm using in each operation. In the example above, $R_1 - \frac {1}{2} R_2$ , the coefficients for $R_2$ would have to come from the second matrix that would follow the first operation $R_2 - 2R_1$. I just asked this because I wasn't really sure how to get rid of that paradox if we were technically allowed to do more than one operation at a time.

Last edited: May 21, 2014