RREF guide without gaussian elimination/faster method

So what is another way to find the RREF without the gaussian elimination, but a much faster method

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So i found a way to find a quick way to find RREF for many matrices. I look at the columns and see if they form an equation or not and if they dont then it must be a pivot column. For example

Lets say this a the matrix

3  0  -18
2  17 -12
5  31 -30


Now instead of doing gaussian elimination and taking forever, look at the matrix carefully. Notice columns 1 does not match with column 2 and column 2 doesnt match with column three. but -6 times column 1 = column 3

so the RREF would be the following

1  0  -6
0  1   0
0  0   0


Now notice the first column and third column and look how it is related to the original matrix without the RREF.

You can also add columns with each other with the appropriate scalars to see if they are related with each other, For example look at this RREF

1  0  5
0  1  3
0  0  0


notice that 5 times column 1 + 3 times column 2 = column 3

So as you can tell this is a much faster/easier way to calculate the RREF.

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Just out of curiosity, did you actually not know the answer and found it within seconds? Or did you already know the answer? – Ataraxia Jul 21 '13 at 15:29
I looked in a book and found a matrix but i didnt know the answer – jyuserersh Jul 21 '13 at 15:33
The question asked for a method. "Look at the matrix and see the answer" is not what I'd call a method. Also, consider what would happen with a 10-by-10 matrix. (Gaussian elimination is routinely used on matrices with thousands of rows and columns.) – Andreas Blass Jul 21 '13 at 15:50
that is true, but for that you use a calculator or a computer to calculate that big of a problem. This is for small matrices. – jyuserersh Jul 21 '13 at 16:37
and if u look at this carefully it looks the column correspondence property – jyuserersh Jul 21 '13 at 16:38