# solve systems of linear equations 
import numpy as np
'''
3x - y = 7
2x + y = 8
----------
5x = 15
x = 3; y = 2
'''

mat = np.array([[3, -1], [2, 1]]) # 2 x 2
rhs = np.array([[7, 8]]) # 1 x 2
out = np.linalg.inv(mat) @ rhs.transpose() # (2 x 2) @ (1 x 2).T
out

array([[3.],
       [2.]])

I = np.eye(3, 3)
I

array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

'''
Ainv @ A = I
'''
mat = np.random.rand(2, 2)
I = np.linalg.inv(mat) @ mat
I

array([[1.00000000e+00, 3.32227679e-16],
       [3.47665962e-18, 1.00000000e+00]])

NOTE: The output matrix does not look exactly what we expect, this is because of the limitation of floating point precision and for practical purposes e-16 can be considered as zero or very close to zero

singular = np.array([[1, 0], [0, 0]])
singular

array([[1, 0],
       [0, 0]])

try:
    np.linalg.inv(singular)
except Exception as e:
    print(f'Cannot find inverse because: {e}')

Cannot find inverse because: Singular matrix

det_singular = np.linalg.det(singular)
det_singular

0.0