[FIXED] Solving System of linear equation using Cramer's method in Python

Issue

I am trying to solve the following system of linear equations:

10x1+ 40x2+ 70x3= 300
20x1+ 50x2+ 80x3= 360
30x1+ 60x2+ 80x3= 390

by using Cramer's method implementing a function by scratch:

def cramer(mat, constant): # takes the matrix and the costants
    
    D = np.linalg.det(mat) # calculating the determinant of the original matrix

# substitution of the column with costant and creating new matrix
    mat1 = np.array([constant, mat[1], mat[2]])
    mat2 = np.array([mat[0], constant, mat[2]])
    mat3 = np.array([mat[0], mat[1], constant])  
    
#calculatin determinant of the matrix
    D1 = np.linalg.det(mat1)
    D2 = np.linalg.det(mat2)
    D3 = np.linalg.det(mat3)
    
#finding the X1, X2, X3
    X1 = D1/D
    X2 = D2/D
    X3 = D3/D
    
    print(X1, X2, X3)

By using the above function on the system

a = np.array([[10, 40, 70],
             [20, 50, 80],
             [30, 60, 80]])

b = np.array([300, 360, 390])

I get the following result:

cramer(a,b)
-22.99999999999996 21.999999999999964 2.999999999999993

I have solved the system using the numpy function np.linalg.solve and I get another result:

x = np.linalg.solve(a, b)
[1. 2. 3.]

I cannot spot the formula error in the function I have witten. What should I adjust in the fuction in order to make it working properly?

Solution

The main problem is how you select the columns of a, i.e. you are actually selecting the rows of a rather than the columns. You can fix it by changing the matrix initializations to this:

    mat1 = np.array([constant, mat[:,1], mat[:,2]])
    mat2 = np.array([mat[:,0], constant, mat[:,2]])
    mat3 = np.array([mat[:,0], mat[:,1], constant])  

Basically mat[:,1] is saying something like mat[all rows, column 1].

Answered By - Andrew Wei