from math import *

global eps

def deter(count, initial_matrix):                                    #computation of determinant
    matrix = [[0 for j in range(count)] for i in range(count)] 
    
    for i in range(count):
        for j in range(count):
            matrix[i][j] = initial_matrix[i][j]
    det = 1
    for k in range(count):                                           #bringing matrix to the 
        for i in range(k, count):                                    #upper-traingular form
            tmp = matrix[i][k]
            if (fabs(tmp) > eps):
                for j in range(count):
                    matrix[i][j] = matrix[i][j] / tmp
                det *= tmp
                
                if (i != k):
                    for j in range(count):
                        matrix[i][j] = matrix[i][j] - matrix[k][j]
                
    for i in range(count):                                           #composition of
        det *= matrix[i][i]                                          #diagonal elements
    return det


def inverse(count, initial_matrix, temporary):                       #computation of inverse matrix
    matrix = [[0 for j in range(count)] for i in range(count)]
    
    for i in range(count):
        for j in range(count):
            matrix[i][j] = initial_matrix[i][j]
    
    for i in range(count):
        for j in range(count):
            if (i == j):
                temporary[i][j] = 1
    
    for k in range(count):                                           #bringing matrix to the 
        for i in range(k, count):                                    #upper-traingular form
            tmp = matrix[i][k]
            if (fabs(tmp) > eps):
                for j in range(count):
                    matrix[i][j] = matrix[i][j] / tmp
                    temporary[i][j] = temporary[i][j] / tmp
                
                if (i != k):
                    for j in range(count):
                        matrix[i][j] = matrix[i][j] - matrix[k][j]
                        temporary[i][j] = temporary[i][j] - temporary[k][j]
  
    
    for k in range(count - 1, -1, -1):                               #bringing matrix to the
        for i in range(count - 1, -1, -1):                           #bottom-traingular form
            tmp = matrix[i][k]

            if (fabs(tmp) > eps):
                for j in range(count):
                    matrix[i][j] = matrix[i][j] / tmp
                    temporary[i][j] = temporary[i][j] / tmp
                
                if (i != k):
                    for j in range(count):
                        matrix[i][j] = matrix[i][j] - matrix[k][j]
                        temporary[i][j] = temporary[i][j] - temporary[k][j]


def conditionality(count, initial_matrix):                           #computation of conditionality
    matrix = [[0 for j in range(count)] for i in range(count)]    
    
    for i in range(count):
        for j in range(count):
            matrix[i][j] = initial_matrix[i][j]
    
    conditionality1 = 0                                              #conditionality of matrix
    for i in range(count):
        summa = 0
        for j in range(count):
            summa += fabs(matrix[i][j])
            
        if (not(i) or summa > conditionality1):
            conditionality1 = summa
    
    
    temporary = [[0 for j in range(count)] for i in range(count)]
    inverse(count, initial_matrix, temporary)
    
    conditionality2 = 0                                              #conditionality of inverse matrix
    for i in range(count):
        summa = 0
        for j in range(count):
            summa += fabs(temporary[i][j])
            
        if (not(i) or summa > conditionality2):
            conditionality2 = summa
            
    
    return conditionality2 * conditionality1




def gauss(count, initial_matrix, x):                                #gauss method
    matrix = [[0 for j in range(count + 1)] for i in range(count)]
    
    for i in range(count):
        for j in range(count + 1):
            matrix[i][j] = initial_matrix[i][j]
    
    for k in range(count):                                          #reduction to trangular form
        for i in range(k, count):
            tmp = matrix[i][k]
            if (fabs(tmp) > eps):
                for j in range(count + 1):
                    matrix[i][j] = matrix[i][j] / tmp
                if (i != k):
                    for j in range(count + 1):
                        matrix[i][j] = matrix[i][j] - matrix[k][j]

    for k in range(count-1, -1, -1):                                #computation answers x[i]
        x[k] = matrix[k][count]
        for i in range(k):
            matrix[i][count] = matrix[i][count] - matrix[i][k] * x[k]
    
    
    
def modified_gauss(count, initial_matrix, x):                       #modified gauss method
    matrix = [[0 for j in range(count + 1)] for i in range(count)]
    
    for i in range(count):
        for j in range(count + 1):
            matrix[i][j] = initial_matrix[i][j]
    
    for k in range(count):
        max_elem = fabs(matrix[k][k])
        max_index = k
        for i in range(k + 1, count):
            if (max_elem < fabs(matrix[i][k])):                     #selection of maximum element
                max_elem = fabs(matrix[i][k])
                max_index = i
        
        if (max_elem < eps):
            return
        
        for j in range(count + 1):
            tmp = matrix[k][j]
            matrix[k][j] = matrix[max_index][j]
            matrix[max_index][j] = tmp
        
        for i in range(k, count):                                    #reduction to trangular form
            tmp = matrix[i][k]
            if (fabs(tmp) > eps):
                for j in range(count + 1):
                    matrix[i][j] = matrix[i][j] / tmp
                
                if (i != k):
                    for j in range(count + 1):
                        matrix[i][j] = matrix[i][j] - matrix[k][j]
                
    
    for k in range(count - 1, -1, -1):                               #computation answers x[i]
        x[k] = matrix[k][count]
        for i in range(k):
            matrix[i][count] = matrix[i][count] - matrix[i][k] * x[k] 
        
        
            
eps = 0.000001
 
print("input matrix range")          
count = int(input())
x = [0 for j in range(count)]
print("if you want input from keyboard, press 1,\nif you want to use formula press 2,\nif you want input from file press 3.")
inp = int(input())
if inp == 1:                                                   #from terminal
    matrix = []
    for i in range(count): 
        print("input", i + 1, "line")
        matrix.append(list(map(float, input().split())))

elif inp == 2:                                                 #using formula
    matrix = [[0 for j in range(count + 1)] for i in range(count)]
    for i in range(count): 
        for j in range(count):
            if (i != j):
                matrix[i][j] = (i + j) / 65
            else:
                matrix[i][j] = 275 + j / 15 + i / 50
        matrix[i][count] = 750 - i * i * i
        
elif inp == 3:                                                 #from file named 'input.txt'
    fin = open('input.txt', 'r')
    matrix = []
    for i in range(count): #schitat matrix
        matrix.append(list(map(float, (fin.readline()).split())))    
    fin.close()

print("\n")
det = deter(count, matrix)
print("determinant D =", det)

if (det == 0):
    print("degenerate matrix\n")

else:
    print("M =", conditionality(count, matrix), "\n")
    
    print("START OF GAUSS METHOD\n")
    gauss(count, matrix, x)
    for i in range(count):
        print("x[" + str(i + 1) + "] = " + str(x[i]))
    print("")
    print("END OF GAUSS METHOD\n")
    
    print("START OF MODIFIED GAUSS METHOD\n")
    modified_gauss(count, matrix, x)
    for i in range(count):
        print("x[" + str(i + 1) + "] = " + str(x[i]))
    print("")
    print("END OF MODIFIED GAUSS METHOD\n")
    
    
    print("INVERSE MATRIX\n")
    temporary = [[0 for j in range(count)] for i in range(count)]
    inverse(count, matrix, temporary)
    for elem in temporary:
        for el in elem:
            print(el, end = "  ")
        print("")
    
    print("\n")