
'''simple docstring'''


def lowercase_	(   _lowercase				)     ->							list[list]:
					'''simple docstring'''

					lowerCamelCase_					:      Optional[Any]     =		current_set.copy()
					for row_index, row in enumerate(_lowercase				):
										lowerCamelCase_					:      List[Any]     =		row[0]
										for column_index, column in enumerate(_lowercase				):
															if magnitude == 0:
																				lowerCamelCase_					:      Any     =		column
																				continue
															lowerCamelCase_					:      Tuple     =		column / magnitude
    # Subtract to cancel term
					lowerCamelCase_					:      Optional[Any]     =		current_set[0]
					lowerCamelCase_					:      List[str]     =		[first_row]
					lowerCamelCase_					:      str     =		current_set[1::]
					for row in current_set:
										lowerCamelCase_					:      List[str]     =		[]
										# If first term is 0, it is already in form we want, so we preserve it
										if row[0] == 0:
															final_set.append(_lowercase				)
															continue
										for column_index in range(len(_lowercase				)				):
															temp_row.append(first_row[column_index] - row[column_index]				)
										final_set.append(_lowercase				)
					# Create next recursion iteration set
					if len(final_set[0]				) != 3:
										lowerCamelCase_					:      str     =		final_set[0]
										lowerCamelCase_					:      Optional[int]     =		[]
										lowerCamelCase_					:      int     =		[]
										for row in final_set[1::]:
															current_first_column.append(row[0]				)
															next_iteration.append(row[1::]				)
										lowerCamelCase_					:      Dict     =		simplify(_lowercase				)
										for i in range(len(_lowercase				)				):
															resultant[i].insert(0			,       current_first_column[i]				)
										resultant.insert(0			,       _lowercase				)
										lowerCamelCase_					:      Tuple     =		resultant
					return final_set




def lowercase_	(   _lowercase				)     ->							list:
					'''simple docstring'''

					if len(_lowercase				) == 0:
										raise IndexError('''solve_simultaneous() requires n lists of length n+1'''				)
					lowerCamelCase_					:      Optional[Any]     =		len(_lowercase				) + 1
					if any(len(_lowercase				) != _length for item in equations				):
										raise IndexError('''solve_simultaneous() requires n lists of length n+1'''				)
					for row in equations:
										if any(not isinstance(_lowercase			,       (int, float)				) for column in row				):
															raise ValueError('''solve_simultaneous() requires lists of integers'''				)
					if len(_lowercase				) == 1:
										return [equations[0][-1] / equations[0][0]]
					lowerCamelCase_					:      Optional[Any]     =		equations.copy()
					if any(0 in row for row in data_set				):
										lowerCamelCase_					:      List[str]     =		data_set.copy()
										lowerCamelCase_					:      Dict     =		[]
										for row_index, row in enumerate(_lowercase				):
															if 0 not in row:
																				lowerCamelCase_					:      List[Any]     =		data_set.pop(_lowercase				)
																				break
										if not full_row:
															raise ValueError('''solve_simultaneous() requires at least 1 full equation'''				)
										data_set.insert(0			,       _lowercase				)
					lowerCamelCase_					:      List[Any]     =		data_set.copy()
					lowerCamelCase_					:      Optional[int]     =		simplify(_lowercase				)
					lowerCamelCase_					:      str     =		simplified[::-1]
					lowerCamelCase_					:      list     =		[]
					for row in simplified:
										lowerCamelCase_					:      Union[str, Any]     =		row[-1]
										if not solutions:
															if row[-2] == 0:
																				solutions.append(0				)
																				continue
															solutions.append(current_solution / row[-2]				)
															continue
										lowerCamelCase_					:      str     =		row.copy()[: len(_lowercase				) - 1 :]
										while temp_row[0] == 0:
															temp_row.pop(0				)
										if len(_lowercase				) == 0:
															solutions.append(0				)
															continue
										lowerCamelCase_					:      int     =		temp_row[1::]
										lowerCamelCase_					:      List[str]     =		temp_row[::-1]
										for column_index, column in enumerate(_lowercase				):
															current_solution -= column * solutions[column_index]
										solutions.append(_lowercase				)
					lowerCamelCase_					:      Optional[Any]     =		[]
					for item in solutions:
										final.append(float(round(_lowercase			,       5				)				)				)
					return final[::-1]


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					__lowercase							:   int					=     [
					    [2, 1, 1, 1, 1, 4],
					    [1, 2, 1, 1, 1, 5],
					    [1, 1, 2, 1, 1, 6],
					    [1, 1, 1, 2, 1, 7],
					    [1, 1, 1, 1, 2, 8],
					]
					print(solve_simultaneous(eq))
					print(solve_simultaneous([[4, 2]]))




'''simple docstring'''


from __future__ import annotations

from decimal import Decimal

from numpy import array




def lowercase_	(   _lowercase				)     ->							list[list[float]]:
					'''simple docstring'''

					lowerCamelCase_					:      Tuple     =		Decimal

					# Check if the provided matrix has 2 rows and 2 columns
					# since this implementation only works for 2x2 matrices
					if len(_lowercase				) == 2 and len(matrix[0]				) == 2 and len(matrix[1]				) == 2:
										# Calculate the determinant of the matrix
										lowerCamelCase_					:      Union[str, Any]     =		float(
										    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
										lowerCamelCase_					:      Optional[Any]     =		[[0.0, 0.0], [0.0, 0.0]]
										lowerCamelCase_,      lowerCamelCase_					:      int     =		matrix[1][1], matrix[0][0]
										lowerCamelCase_,      lowerCamelCase_					:      Dict     =		-matrix[1][0], -matrix[0][1]

										# Calculate the inverse of the matrix
										return [
										    [(float(d(_lowercase				)				) / determinant) or 0.0 for n in row] for row in swapped_matrix
										]
					elif (
					    len(_lowercase				) == 3
					    and len(matrix[0]				) == 3
					    and len(matrix[1]				) == 3
					    and len(matrix[2]				) == 3
					):
										# Calculate the determinant of the matrix using Sarrus rule
										lowerCamelCase_					:      Tuple     =		float(
										    (
										        (d(matrix[0][0]				) * d(matrix[1][1]				) * d(matrix[2][2]				))
										        + (d(matrix[0][1]				) * d(matrix[1][2]				) * d(matrix[2][0]				))
										        + (d(matrix[0][2]				) * d(matrix[1][0]				) * d(matrix[2][1]				))
										    )
										    - (
										        (d(matrix[0][2]				) * d(matrix[1][1]				) * d(matrix[2][0]				))
										        + (d(matrix[0][1]				) * d(matrix[1][0]				) * d(matrix[2][2]				))
										        + (d(matrix[0][0]				) * d(matrix[1][2]				) * d(matrix[2][1]				))
										    )				)
										if determinant == 0:
															raise ValueError('''This matrix has no inverse.'''				)

										# Creating cofactor matrix
										lowerCamelCase_					:      Any     =		[
										    [d(0.0				), d(0.0				), d(0.0				)],
										    [d(0.0				), d(0.0				), d(0.0				)],
										    [d(0.0				), d(0.0				), d(0.0				)],
										]
										lowerCamelCase_					:      Union[str, Any]     =		(d(matrix[1][1]				) * d(matrix[2][2]				)) - (
										    d(matrix[1][2]				) * d(matrix[2][1]				)
										)
										lowerCamelCase_					:      Any     =		-(
										    (d(matrix[1][0]				) * d(matrix[2][2]				)) - (d(matrix[1][2]				) * d(matrix[2][0]				))
										)
										lowerCamelCase_					:      Tuple     =		(d(matrix[1][0]				) * d(matrix[2][1]				)) - (
										    d(matrix[1][1]				) * d(matrix[2][0]				)
										)
										lowerCamelCase_					:      Optional[Any]     =		-(
										    (d(matrix[0][1]				) * d(matrix[2][2]				)) - (d(matrix[0][2]				) * d(matrix[2][1]				))
										)
										lowerCamelCase_					:      Tuple     =		(d(matrix[0][0]				) * d(matrix[2][2]				)) - (
										    d(matrix[0][2]				) * d(matrix[2][0]				)
										)
										lowerCamelCase_					:      List[Any]     =		-(
										    (d(matrix[0][0]				) * d(matrix[2][1]				)) - (d(matrix[0][1]				) * d(matrix[2][0]				))
										)
										lowerCamelCase_					:      Any     =		(d(matrix[0][1]				) * d(matrix[1][2]				)) - (
										    d(matrix[0][2]				) * d(matrix[1][1]				)
										)
										lowerCamelCase_					:      Tuple     =		-(
										    (d(matrix[0][0]				) * d(matrix[1][2]				)) - (d(matrix[0][2]				) * d(matrix[1][0]				))
										)
										lowerCamelCase_					:      str     =		(d(matrix[0][0]				) * d(matrix[1][1]				)) - (
										    d(matrix[0][1]				) * d(matrix[1][0]				)
										)

										# Transpose the cofactor matrix (Adjoint matrix)
										lowerCamelCase_					:      str     =		array(_lowercase				)
										for i in range(3				):
															for j in range(3				):
																				lowerCamelCase_					:      int     =		cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
										lowerCamelCase_					:      int     =		array(_lowercase				)
										for i in range(3				):
															for j in range(3				):
																				inverse_matrix[i][j] /= d(_lowercase				)

        # Calculate the inverse of the matrix
										return [[float(d(_lowercase				)				) or 0.0 for n in row] for row in inverse_matrix]
					raise ValueError('''Please provide a matrix of size 2x2 or 3x3.'''				)


