gj4, Python 3 - rextester

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# augmented matrix
matrix = [[3, 2, 3, 4, 2, 1, 0, 0, 0, 0],   
          [2, 3, 4, 2, 1, 0, 1, 0, 0, 0],
          [3, 1, 2, 5, 2, 0, 0, 1, 0, 0],
          [1, 3, 1, 4, 1, 0, 0, 0, 1, 0],
          [3, 2, 2, 3, 1, 0, 0, 0, 0, 1]]

def matrix_elimination(matrix):
    # pollaplasiazoume ta stoixeia 1hs grammis me ton antistrofo tou protou stoixeiou
    # afairoume tin 1h grammi apo ti 2h (elimination)
    # epanalamvanoume tin parapano diadikasia,
    # oste na afairethei h 1h grammi apo oles
    # epanalambanetai h idia diadikasia gia ti 2h grammi
    i, k, m, n = 0, 0, 0, 5
    while i < n:    # o vroxos aftos pollaplasiazei ta stoixeia kathe grammis me ton antistrofo 1ou, 2ou k.o.k stoixeiou analoga me ti grammi pou vriskomaste
        j = 0               # i -> diasxizei tis grammes
        var = matrix[i][i]  # j -> diasxizei ta stoixeia kathe grammis
#        print ("var =", var, "", "i =", i)
        for e in matrix[i]: 
           if var != 0:
            e = e * (1 / var)
            matrix[i][j] = e
            j = j + 1
 #       print ("matrix = ", matrix)    
        m = k + 1
        while m < n:    # o vroxos aftos pollaplasiazei ta stoixeia mias grammis me tin "var1" kai ektelei afairesi ton grammon
            var1 = matrix[m][i]
#            print ("var1 =", var1)
            k = i       
            while k < 2*n:  # k -> diasxizei ola ta stoixeia mias grammis
                matrix[m][k] = matrix[m][k] - (matrix[i][k] * var1 )
                k = k + 1    
            m = m + 1
        k = i
        k = k + 1      
        i = i + 1
#        print (*matrix, sep = "\n")    
    return matrix

upper_triangular_matrix = matrix_elimination(matrix)
print ("upper_triangular_matrix =",*upper_triangular_matrix, sep = "\n")          
 
def upper_triangular_matrix_elimination(upper_triangular_matrix):
    n = 5
    i, j, k = n-1, n - 1, 0
#   pollaplasiazoume tin teleftea grammi me to telefteo stoixeio tis protelefteas kai afairoume "proteleftea - teleftea"...
#   anaferomaste sto proto miso tou arxikou pinaka
    while j > 0:
      
#      print ("j =", j)
      i = n - 1 - k  
      while i > 0:
         p = 0
         var = upper_triangular_matrix[i - 1][j]
#        print ("var =", var, "", "i =", i)
         for e in upper_triangular_matrix[j]: 
             e = e * var
#             if e != 0:
#                break
             upper_triangular_matrix[i - 1][p] = upper_triangular_matrix[i - 1][p] - e
             p = p + 1
#        print (*inverse_matrix, sep = "\n")
         i = i - 1
      k = k + 1
      j = j - 1           
    return upper_triangular_matrix
diagonal_matrix = upper_triangular_matrix_elimination(upper_triangular_matrix)                      
print ("diagonal_matrix =", *diagonal_matrix, sep = "\n")

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Absolute running time: 0.23 sec, cpu time: 0.21 sec, memory peak: 6 Mb, absolute service time: 0,24 sec