Scott, it seems to have bug in your code:
> A=matrix(c(1,-2,3,9,-1,3,0,-4,2,-5,5,17),ncol=4,byrow=T)
> ref(A)
[,1] [,2] [,3] [,4]
[1,] 1 -2.5 2.166667 7.833333
[2,] 0 1.0 3.000000 5.000000
[3,] 0 0.0 1.000000 2.000000
the row echelon is apparently incorrect.
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