2.7

import numpy as np

A = np.array([[4, 2, -1], [3, -1, 2], [11, 3, 0]])
b = np.array([2, 10, 8])
x = np.linalg.lstsq(A, b)
if np.linalg.matrix_rank(A) == np.linalg.matrix_rank(np.column_stack((A, b))):
if np.linalg.matrix_rank(A) == A.shape[1]:
print("线性方程组有唯一解")
x_unique = np.linalg.solve(A, b)
print("唯一解 x =", x_unique)
else:
print("线性方程组有无穷多解")
x_least_squares = np.linalg.lstsq(A, b, rcond=None)[0]
print("最小二乘解 x =", x_least_squares)
else:
print("线性方程组无解")
x_least_squares = np.linalg.lstsq(A, b, rcond=None)[0]
print("最小二乘解 x =", x_least_squares)
x_min_norm = np.linalg.pinv(A).dot(b)
print("最小范数解 x =", x_min_norm)

A = np.array([[2, 3, 1], [1, -2, 4], [3, 8, -2],[4,-1,9]])
b = np.array([4, -5, 13,-6])

x = np.linalg.lstsq(A, b)

if np.linalg.matrix_rank(A) == np.linalg.matrix_rank(np.column_stack((A, b))):
if np.linalg.matrix_rank(A) == A.shape[1]:
print("线性方程组有唯一解")
x_unique = np.linalg.solve(A, b)
print("唯一解 x =", x_unique)
else:
print("线性方程组有无穷多解")
x_least_squares = np.linalg.lstsq(A, b, rcond=None)[0]
print("最小二乘解 x =", x_least_squares)
else:
print("线性方程组无解")
x_least_squares = np.linalg.lstsq(A, b, rcond=None)[0]
print("最小二乘解 x =", x_least_squares)
x_min_norm = np.linalg.pinv(A).dot(b)
print("最小范数解 x =", x_min_norm)