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from decimal import Decimal

from typing import List

def inverse_of_matrix(matrix: List[List[float]]) -> List[List[float]]:

"""

A matrix multiplied with its inverse gives the identity matrix.

This function finds the inverse of a 2x2 matrix.

If the determinant of a matrix is 0, its inverse does not exist.

Sources for fixing inaccurate float arithmetic:

https://stackoverflow.com/questions/6563058/how-do-i-use-accurate-float-arithmetic-in-python

https://docs.python.org/3/library/decimal.html

>>> inverse_of_matrix([[2, 5], [2, 0]])

[[0.0, 0.5], [0.2, -0.2]]

>>> inverse_of_matrix([[2.5, 5], [1, 2]])

Traceback (most recent call last):

...

ValueError: This matrix has no inverse.

>>> inverse_of_matrix([[12, -16], [-9, 0]])

[[0.0, -0.1111111111111111], [-0.0625, -0.08333333333333333]]

>>> inverse_of_matrix([[12, 3], [16, 8]])

[[0.16666666666666666, -0.0625], [-0.3333333333333333, 0.25]]

>>> inverse_of_matrix([[10, 5], [3, 2.5]])

[[0.25, -0.5], [-0.3, 1.0]]

"""

D = Decimal # An abbreviation to be conciseness

# Calculate the determinant of the matrix

determinant = D(matrix[0][0]) * D(matrix[1][1]) - D(matrix[1][0]) * D(matrix[0][1])

if determinant == 0:

raise ValueError("This matrix has no inverse.")

# Creates a copy of the matrix with swapped positions of the elements

swapped_matrix = [[0.0, 0.0], [0.0, 0.0]]

swapped_matrix[0][0], swapped_matrix[1][1] = matrix[1][1], matrix[0][0]

swapped_matrix[1][0], swapped_matrix[0][1] = -matrix[1][0], -matrix[0][1]

# Calculate the inverse of the matrix

return [[float(D(n) / determinant) or 0.0 for n in row] for row in swapped_matrix]

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