import numpy as np

def eliminateCol$($A$,$ k$=0)$:

# guarantee that A is float type

if A$.$dtype.kind $!=\text{'}$f$\text{'}$ and A$.$dtype.kind $!=\text{'}$c$\text{'}$: return None

# load dimensions

dim$0,$dim$1=$ np$.$shape$($A$)$

# initialize outputs

swap $=$ np$.$identity$($dim$0,$dtype$=$float$)$

oper $=$ np$.$identity$($dim$0,$dtype$=$float$)$

# k $<$ dim$1$

k $=$ k$\%$dim$1$

# if the column k is zero column, return

if A$[$:$,$k$].$any$()==$ False: return swap,oper

i$0=0$

if k $!=0$:

# find a row i$0$ whose left most k$-1$ entries are all zero

while i$0<$ dim$0$:

if A$[$i$0,0$:k$].$any$()==$ False: break

i$0+=1$

# find a nonzero element A$[$i$,$k$]$ to use as a pivot

i $=$ i$0$;

while i $<$ dim$0$:

if A$[$i$,$k$]!=0$: break

i$+=1$

if i $==$ dim$0$: return swap,oper

# swap the row i with the row i$0$

if i $!=$ i$0$: A$[[$i$0,$i$],]=$ A$[[$i$,$i$0],]$;swap$[[$i$0,$i$],]=$ swap$[[$i$,$i$0],]$

# row i$0$ is the pivot row and A$[$j$,$k$]$ is zero for i$0=0$:

if A$[$i$,$j$]!=0$:

quot $=$ A$[$i$,$j$]/$ A$[$k$,$j$]$

A$[$i$,$:$]=$ A$[$i$,$:$]-$ quot$*$A$[$k$,$:$]$

oper$[$i$,$k$]=-$quot

i$-=1$

return oper

def eliminateScale$($A$)$:

# guarantee that A is float type

if A$.$dtype.kind $!=\text{'}$f$\text{'}$ and A$.$dtype.kind $!=\text{'}$c$\text{'}$: return None

# load dimensions

dim$0,$dim$1=$ np$.$shape$($A$)$

# initialize outputs

oper $=$ np$.$identity$($dim$0,$dtype$=$float$)$

# find pivots

i $=0$

while i $<$ dim$0$:

j $=0$

while j $<$ dim$1$:

if A$[$i$,$j$]==0$: j$+=1$; continue

else:

oper$[$i$,$i$]=1/$A$[$i$,$j$]$

A$[$i$,$j:$]=$ A$[$i$,$j:$]/$A$[$i$,$j$]$

break

j$+=1$

i$+=1$

return oper

def echelon$\_$to$\_$rref $($A$)$:

# Perform Gaussian elimination on matrix A

for k in range$($A$.$shape$[1])$:

eliminateCol$($A$,$ k$)$

eliminateRow$($A$,$ k$)$

# Finally, scale the matrix

eliminateScale$($A$)$

return A

# Example Usage:

A $=$ np$.$array$([[0\mathrm{.}0,-1\mathrm{.}0,2\mathrm{.}0,8\mathrm{.}0],$

$[0\mathrm{.}0,0\mathrm{.}0,-1\mathrm{.}0,-11\mathrm{.}0],$

$[0\mathrm{.}0,2\mathrm{.}0,-1\mathrm{.}0,-3\mathrm{.}0]],$ dtype$=$float$)$

reduced $=$ echelon$\_$to$\_$rref$($A$.$copy$())$

print$($A$)$

print$($reduced$)$

Reduced row echelon form A matrix is false. Rref of A should be

$[[0,1,0,14],$

$[0,0,1,11],$

$[0,0,0,-20]]$

System should be inconsistent. Fix the code.