I have created a function determinant which outputs a determinant of a 3x3 matrix. I also need to create a function to invert that matrix however the code doesn't seem to work and I can't figure out why.
M = np.array([
[4.,3.,9.],
[2.,1.,8.],
[10.,7.,5.]
])
def inverse(M):
'''
This function finds the inverse of a matrix using the Cramers rule.
Input: Matrix - M
Output: The inverse of the Matrix - M.
'''
d = determinant(M) # Simply returns the determinant of the matrix M.
counter = 1
array = []
for line in M: # This for loop simply creates a co-factor of Matrix M and puts it in a list.
y = []
for item in line:
if counter %2 == 0:
x = -item
else:
x = item
counter += 1
y.append(x)
array.append(y)
cf = np.matrix(array) # Translating the list into a matrix.
adj = np.matrix.transpose(cf) # Transposing the matrix.
inv = (1/d) * adj
return inv
OUTPUT:
via inverse(M):
[[ 0.0952381 -0.04761905 0.23809524],
[-0.07142857 0.02380952 -0.16666667],
[ 0.21428571 -0.19047619 0.11904762]]
via built-in numpy inverse function:
[[-1.21428571 1.14285714 0.35714286]
[ 1.66666667 -1.66666667 -0.33333333]
[ 0.0952381 0.04761905 -0.04761905]]
As you can see some of the numbers match and I'm just not sure why the answer isn't exact as I'm using the formula correctly.
You co-factor matrix calculation isn't correct.
def inverse(M):
d = np.linalg.det(M)
cf_mat = []
for i in range(M.shape[0]):
for j in range(M.shape[1]):
# for each position we need to calculate det
# of submatrix without current row and column
# and multiply it on position coefficient
coef = (-1) ** (i + j)
new_mat = []
for i1 in range(M.shape[0]):
for j1 in range(M.shape[1]):
if i1 != i and j1 != j:
new_mat.append(M[i1, j1])
new_mat = np.array(new_mat).reshape(
(M.shape[0] - 1, M.shape[1] - 1))
new_mat_det = np.linalg.det(new_mat)
cf_mat.append(new_mat_det * coef)
cf_mat = np.array(cf_mat).reshape(M.shape)
adj = np.matrix.transpose(cf_mat)
inv = (1 / d) * adj
return inv
This code isn't very effective, but here you can see, how it should be calculated. More information and clear formula you can find at Wiki.
Output matrix:
[[-1.21428571 1.14285714 0.35714286]
[ 1.66666667 -1.66666667 -0.33333333]
[ 0.0952381 0.04761905 -0.04761905]]