I added to the code linked above and made it into an @interact that takes one all the way to reduced echelon form:
(a formatting issue is having one of the @interacts not in the code box, and messing with the indentation a bit. Be sure to copy-paste the hanging @interact and indent in the nested one.)

@interact
def _(row = slider([1..10], label='rows', default = 3) , col = slider([1..10], label='columns', default = 3)):

```
@interact(layout=dict(top=[['M', 'b']]))
def gauss_method(b= random_matrix(QQ,nrows = row, ncols = 1, algorithm='echelonizable', rank = 1), M=random_matrix(QQ,nrows = row, ncols = col, algorithm='echelonizable', rank = min(row,col)),rescale_leading_entry=True ):
M=M.augment(b, subdivide = True)
N=copy(M)
N=N.rref()
num_rows=M.nrows()
num_cols=M.ncols()
print("Reduced echelon form:")
show(N)
print("Steps")
show(M)
col = 0 # all cols before this are already done
for row in range(0,num_rows):
# ?Need to swap in a nonzero entry from below
while (col < num_cols
and M[row][col] == 0):
for i in M.nonzero_positions_in_column(col):
if i > row:
print " R",row+1,"\;\circlearrowright \; R",i+1
M.swap_rows(row,i)
show(M)
break
else:
col += 1
if col >= num_cols:
break
# Now guaranteed M[row][col] != 0
if (rescale_leading_entry
and M[row][col] != 1):
print "",1/M[row][col],"R",row+1
M.rescale_row(row,1/M[row][col])
show(M)
change_flag=False
for changed_row in range(row+1,num_rows):
if M[changed_row][col] != 0:
change_flag=True
factor=-1*M[changed_row][col]/M[row][col]
print "",factor," R",changed_row+1,"+R",changed_row
M.add_multiple_of_row(changed_row,row,factor)
if change_flag:
show(M)
col +=1
print "Above is the echelon form, let's keep cruising to get the reduced echelon form"
for changed_row in range(1,num_rows):
for row in range(0,changed_row):
factor = -1*M[row][changed_row]
print "",factor," R",changed_row+1,"+R",row+1
M.add_multiple_of_row(row,changed_row,factor)
show(M)
```

Not except by hand. We have many requests for steps in basic algebra and linear algebra and calculus, but it's unlikely that anyone will step up and implement this from scratch. You can do the steps yourself, of course, for pedagogical purposes, but Sage won't show those intermediate ones a priori. Unless you want to implement this :) which would be welcomed, for sure.

Thanks for the reply. It is for pedagogical purposes. I'll think about implementing it.