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Let's say I have a matrix A of arbitrary size, and I perform a finite number of both elementary row/column operations on it, obtaining matrix B.

Are there any unique properties of matrix B that would be the same as matrix A? Such that I will be able to recognize that matrix B is a result of these operations performed on matrix A.

Here are some of the stuff that I've managed to find so far:

From Wikipedia, "Elementary row operations do not change the kernel of a matrix" "Elementary column operations do not change the image, but they do change the kernel."

And I've been told on StackOverflow that "if you don't consider scaling a row/column as an elementary operation a lot more structure is maintained".

Thank you.

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Cross-posted here, right? –  draks ... Aug 29 '12 at 11:54
    
Yup, I was advised to ask here instead. –  user1632240 Aug 29 '12 at 22:10

2 Answers 2

You will not be able to recognize that matrix $B$ originated from matrix $A$. There are other matrices that can be reduced to matrix $B$ via performing elementary operations.

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If I rephrased my question slightly, instead of recognizing that B originated from A, but for Matrix A and B to still share some common un-changed properties as a result of those elementary operations, will there still be an answer? –  user1632240 Aug 29 '12 at 22:16
    
I think all other relevant properties preserved by elementary operations are listed in @Gerry Myerson's answer. –  Godot Aug 30 '12 at 10:49

I think the only things you can count on are that you won't change the number of rows, the number of columns, the rank, or the nullity. If it's a square matrix, you won't change its invertibility.

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