# A very basic question about matrix operations

#### 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?

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#### SammyS

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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 operations at a time. What is it that I'm missing?

#### Regtic

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

#### SammyS

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I edited it and forgot to fix the the plurality. Sorry.
Don't you mean that you teacher never does more than one row operation at a time ?

#### Regtic

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

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

#### Regtic

I guess that explains that. LOL

#### 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

#### SammyS

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

#### Regtic

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

#### SammyS

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

#### Ray Vickson

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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?
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
$$\pmatrix{1 & -1/2 \\-2 & 1}$$

#### Regtic

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
$$\pmatrix{1 & -1/2 \\-2 & 1}$$
So I just can't modify a row that I'm adding to another one?

#### Ray Vickson

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So I just can't modify a row that I'm adding to another one?
It would depend on the order in which you do things, and whether or not you overwrite old entries with new ones, etc.

#### Regtic

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

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"A very basic question about matrix operations"

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