Do Row Operations Maintain Equivalence in Matrix Subtraction?

[MathewsMD]

If you have two arbitrary matrices, A and B, I was wondering if row operations can be performed in any order to produce the same results.

For example, you perform elementary row operations on A to produce A', then do A' - B, then also produce a new matrix through elementary row operations on this new matrix to produce a new matrix C.

Can matrix C still be obtained by doing A (not A') - B, and then performing row operations? By performing row operations, the matrix is still remained intact, correct? There's no distortion of the row space, right?

 

[Mark44]

If you have two arbitrary matrices, A and B, I was wondering if row operations can be performed in any order to produce the same results.

Let's limit the discussion to a single matrix A. You can do elementary row operations in any order. At each step along the way, the new matrix will be equivalent to the one you started with.

For example, you perform elementary row operations on A to produce A', then do A' - B

Why? When you perform an elementary row operation on A, you get a new matrix A' that is equivalent to A, but not equal to it. Some of the entries in the new matrix are different from those in A.

The basic idea is that if Ax = 0, for instance, then A'x = 0 as well, even though A $\neq$ A'. Subtracting one matrix from another is not an elementary row operation, so I don't get the point of your question.

, then also produce a new matrix through elementary row operations on this new matrix to produce a new matrix C.

Can matrix C still be obtained by doing A (not A') - B, and then performing row operations? By performing row operations, the matrix is still remained intact, correct? There's no distortion of the row space, right?

 

[MathewsMD]

Let's limit the discussion to a single matrix A. You can do elementary row operations in any order. At each step along the way, the new matrix will be equivalent to the one you started with.
Why? When you perform an elementary row operation on A, you get a new matrix A' that is equivalent to A, but not equal to it. Some of the entries in the new matrix are different from those in A.

The basic idea is that if Ax = 0, for instance, then A'x = 0 as well, even though A $\neq$ A'. Subtracting one matrix from another is not an elementary row operation, so I don't get the point of your question.

Okay, thank you for the response. I was trying to get at the this question specifically: A' is a reduced form of A, then is (A' + B)' = (A + B)', where (A+B)' is some reduction of A+B. Does the above equation hold true? Note: the reductions on each matrix may involve completely different operations, there's just some kind of row operation being done.

 

[Mark44]

Okay, thank you for the response. I was trying to get at the this question specifically: A' is a reduced form of A, then is (A' + B)' = (A + B)', where (A+B)' is some reduction of A+B. Does the above equation hold true? Note: the reductions on each matrix may involve completely different operations, there's just some kind of row operation being done.

Why don't you try a couple of simple examples - say 2 x 2 matrices or 3 x 3 matrices?

What you wrote is, I think, garbled.

(A' + B)' = (A + B)'

Did you mean (A' + B') = (A + B)'?

Also, by "=" do you mean "is row equivalent to" or "equals"? A professor I had in a 400-level linear algebra class was always very careful to write $\equiv$ when he was doing row operations, a habit that I've followed ever since.

 

[MathewsMD]

Why don't you try a couple of simple examples - say 2 x 2 matrices or 3 x 3 matrices?

What you wrote is, I think, garbled.

Did you mean (A' + B') = (A + B)'?

Also, by "=" do you mean "is row equivalent to" or "equals"? A professor I had in a 400-level linear algebra class was always very careful to write $\equiv$ when he was doing row operations, a habit that I've followed ever since.

Sorry, I meant "equals to." By " ' " I am just referring to any random sequence of elementary row operations (I am not sure if there is better notation). And the ' used for one matrix isn't necessarily the same exact sequence of row operations done on the other matrices.

The only case I can necessarily find where it is not true is when A = B, since then A - B = 0, and no row operations can be used to derive A' - B. I don't seem to be able to prove if it's true for the nontrivial case, though. It does seem to stand, though.

 

[Mark44]

The only case I can necessarily find where it is not true is when A = B, since then A - B = 0

Did you try it with a few specific examples, as I suggested? I don't think the statement is true at all.

, and no row operations can be used to derive A' - B. I don't seem to be able to prove if it's true for the nontrivial case, though. It does seem to stand, though.