Finding the Null Space

Question

The function calculates the RREF and null space of a given matrix with the exception of one case; where the matrix is already in RREF.

def r_r_e_f(matrix):
      a=matrix.copy()
      m, n=numpy.shape(a)
      i=0
      j=0
      rank=0
      l=[]
      while i<=(m-1) and j<=(n-1):
          p=numpy.max(numpy.abs(a[i:m, j]))
          max_p=numpy.argmax(numpy.abs(a[i:m, j]))
          k=max_p+i
          if p==0 or p<=0.0000000000000000001:
              j=j+1
          else:
              rank=rank+1
              c=a.copy()
              c[k, :]=a[i, :]
              c[i, :]=a[k, :]
              a=c.copy()
              a[i, :]=a[i, :]/a[i, j]
              l.append(j)
              for x in range(0, m):
                  if x!=i:
                      a[x, :]=a[x, :]-a[x, j]*a[i, :]
          i=i+1
          j=j+1
      ncols=[]
      for col in range(n):
        if col not in l:
          ncols.append(col)
      nbasis=[]
      for col in ncols:
          basis=numpy.zeros(n)
          basis[col]=1
          for row in range(len(l)):
              pcol=l[row]
              basis[pcol]=-a[row, col]
          nbasis.append(basis)
      return a, nbasis
matrix=numpy.array([[1,0,3,0],[0,0,2,1]], dtype=float)
rref, null_space=r_r_e_f(matrix)
print(matrix)
print("RREF of the matrix: ")
print(rref)
print("basis of null space of the matrix: ")
formatm=[]
if null_space==[]:
    print("null space is a 0 vector")
else:
  for bv in null_space:
      reformat=bv.reshape(-1, 1)
      formatm.append(reformat)
  for i in formatm:
      print(i)

When the matrix provided is already in RREF, the program does not return the correct basis of null space. Can someone please help me identify the problem.`

Here is the output for a matrix that is already in RREF

Matrix:

[[1. 0. 3. 0.]
 [0. 0. 2. 1.]]
RREF of the matrix: 
[[1. 0. 3. 0.]
 [0. 0. 2. 1.]]
basis of null space of the matrix: 
[[-0.]
 [ 1.]
 [ 0.]
 [ 0.]]
[[-3.]
 [ 0.]
 [ 1.]
 [ 0.]]
[[-0.]
 [ 0.]
 [ 0.]
 [ 1.]]

The correct output should return the vectors: [0, 1, 0. 0] and [1.5, 0, -0.5, 1]

However for a matrix which is not in RREF, the output is as expected: Matrix: [[1,2,3,0],[0,2,2,1]] basis of null space of the matrix:

[[-1.]
 [-1.]
 [ 1.]
 [ 0.]]
[[ 1. ]
 [-0.5]
 [ 0. ]
 [ 1. ]]