# Fractional number row reduced echelon form

There is a matrix that we try to find its rref. In its rref, some floating point numbers exist. My question is that all denominators of floating point numbers in rref matrix are same always? Can be there different something like?

[1/2,0,0,1]
[0,0,0,3/5]


I use the information to multiple each row the denominator to get minimum integer.

I assume all denominators are 2 for the example. Even though, I multiply by 2 each row, I don't understand how the result is obtained, the integer one.

It is original matrix, Fractional number rref, Integer numbered rref

[ 1  0  0  0  0  1  0  0  0  0  0  0  0  0]
[ 0  1  0  0  0  1  0  0  1  0  0 -1 -1  1]
[ 0  0  1  0  0  0  0  0  0  0  0  0  1  0]
[ 0  0  0  1  0  1  0  0  1  0  0  0  0  1]
[ 0  0  0  0  1  0  0  0  0  0  0  1  1 -1]
[ 0  0  0  0  0  2  0  0  0  0  0 -1  0  1]
[ 0  0  0  0  0  0  1  0  0  0  0  1  0  0]
[ 0  0  0  0  0  0  0  1  1  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  2  0  0  0 -1  1]
[ 0  0  0  0  0  0  0  0  0  1  0 -1 -1  1]
[ 0  0  0  0  0  0  0  0  0  0  1  0  0  1]

• If the second black matrix is indeed the rref of the first one, you cannot get the integer one. The rref is unique. The last one is not rref, since the leading entry of each row must be $1$. Jan 23, 2018 at 10:15