Is simultaneous row operation allowed in reducing a matrix to row echelon form?

[MathewsMD]

For any given 2 x 3 matrix,

Why are only elementary steps allowed (i.e. aR1, R1 +/- R2, R1 <--> R2) and not any other operation (e.g. 3R1 + 2R2) when reducing the matrix to row echelon form?

Also, is the operation R1 + R2 and R2 + R1 allowed simultaneously? I realize this would just output the exact same equation, which isn't incredibly useful, but is it allowed?

 

[Doug Huffman]

The operations plus and times are closed over linear spaces that are mapped by matrices.

 

[Mark44]

_j

For any given 2 x 3 matrix,

Why are only elementary steps allowed (i.e. aR1, R1 +/- R2, R1 <--> R2) and not any other operation (e.g. 3R1 + 2R2) when reducing the matrix to row echelon form?

The way I've usually seen these elementary row operations is like so (http://en.wikipedia.org/wiki/Elementary_matrix#Operations):
R_i <--> R_j Switch two rows
R_i <-- kR_i (k a nonzero scalar) Replace a row with a nonzero multiple of itself
R_i <-- R_i +kR_j (k a nonzero scalar) Replace a row by itself plus a nonzero multiple of another row

3R₁ + 2R₂ could be effected by replacing R₁ by itself plus 2/3 R₂, followed by replacing R₁ by 3 times itself.

Also, is the operation R1 + R2 and R2 + R1 allowed simultaneously?

I don't understand what your notation means. Which row gets changed?

I realize this would just output the exact same equation, which isn't incredibly useful, but is it allowed?

 

[HallsofIvy]

If you apply the "row operations", "multiply a row by a constant", "swap two rows", and "add a multiple of one row to another" to the identity matrix, you get an "elementary matrix". Applying that row operation to any matrix, A, is the same as multiplying the corresponding elementary matrix by A. For example, if you "add 4 times the first row to the third row" of the identity matrix you get
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1\end{bmatrix}$$
and adding four times the first row to the second row of
$$\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$
gives
$$\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31}+ 4a_{11} & a_{32}+ 4a_{12} & a_{33}+ a_{13} \end{bmatrix}$$

That is exactly the same as
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1\end{bmatrix}\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}$$

In other words, those row operations are exactly the same as multiplying by matrices. If I apply row operations to reduce matrix A to the identity matrix, then multiplying together the corresponding matrices would give the inverse matrix. Applying those row operations to the identity matrix is the same as multiplying all those elementary matrices together, which gives the inverse matrix.

 

[WWGD]

For any given 2 x 3 matrix,

Why are only elementary steps allowed (i.e. aR1, R1 +/- R2, R1 <--> R2) and not any other operation (e.g. 3R1 + 2R2) when reducing the matrix to row echelon form?

Also, is the operation R1 + R2 and R2 + R1 allowed simultaneously? I realize this would just output the exact same equation, which isn't incredibly useful, but is it allowed?

Because these operations are the only ones that preserve the solution to the system. This is obvious for swaps and scaling, but a bit harder for linear combinations. It is also hard (at least for me ; can't think of a way of doing it) to show these are the only operations with those properties.
Still, you can get $3R_1+2R_2$ as a combination of elementary operations.

 

[Erland]

Also, is the operation R1 + R2 and R2 + R1 allowed simultaneously? I realize this would just output the exact same equation, which isn't incredibly useful, but is it allowed?

No, the operations
1. R1 <-- R1 + R2
2. R2 <-- R2 + R1
(I assume that this was what you meant)
cannot be performed simultaneously.

This is easily seen for the matrix

$$\begin{bmatrix}1 & 1 \\ -1 & -1 \end{bmatrix}$$
which is then transformed into the zero matrix

$$\begin{bmatrix}0 & 0 \\ 0& 0 \end{bmatrix}$$
and these two matrices are clearly not row equivalent: the corresponding homogeneous systems have different solution sets, the latter is satisfied by any pair (x,y), which the first one is not.

So, row operations must in principle be performed sequentially. But in many cases, the operations are independent of the operations immediately before, and then they can be performed simultaneously, for example:

1. R2 <-- R2 + R1
2. R3 <-- R3 + R1

Since 1 does not change R1, 1 and 2 can be performed simultaneously. This is a difference from the previous case.