from __future__ import annotations

class 					_lowercase    :






    '''simple docstring'''




    def __init__(    self       ,						lowerCamelCase__ ):
       lowerCAmelCase_:							str       =	TypeError(
           "Matrices must be formed from a list of zero or more lists containing at "
           "least one and the same number of values, each of which must be of type "
           "int or float." )
       if len(lowerCamelCase__ ) != 0:
          lowerCAmelCase_:							int       =	len(rows[0] )
          if cols == 0:
             raise error
          for row in rows:
             if len(lowerCamelCase__ ) != cols:
                raise error
             for value in row:
                if not isinstance(lowerCamelCase__       ,						(int, float) ):
                   raise error
          lowerCAmelCase_:							List[str]       =	rows
       else:
          lowerCAmelCase_:							Optional[Any]       =	[]




    def      _a    (    self ):
       return [[row[i] for row in self.rows] for i in range(len(self.rows[0] ) )]




    @property
    def      _a    (    self ):
       return len(self.rows )




    @property
    def      _a    (    self ):
       return len(self.rows[0] )




    @property
    def      _a    (    self ):
       return (self.num_rows, self.num_columns)




    @property
    def      _a    (    self ):
       return self.order[0] == self.order[1]




    def      _a    (    self ):
       lowerCAmelCase_:							Tuple       =	[
           [0 if column_num != row_num else 1 for column_num in range(self.num_rows )]
           for row_num in range(self.num_rows )
       ]
       return Matrix(lowerCamelCase__ )




    def      _a    (    self ):
       if not self.is_square:
          return 0
       if self.order == (0, 0):
          return 1
       if self.order == (1, 1):
          return int(self.rows[0][0] )
       if self.order == (2, 2):
          return int(
              (self.rows[0][0] * self.rows[1][1])
              - (self.rows[0][1] * self.rows[1][0]) )
       else:
          return sum(
              self.rows[0][column] * self.cofactors().rows[0][column]
              for column in range(self.num_columns ) )




    def      _a    (    self ):
       return bool(self.determinant() )




    def      _a    (    self       ,						lowerCamelCase__       ,						lowerCamelCase__ ):
       lowerCAmelCase_:							Tuple       =	[
           [
               self.rows[other_row][other_column]
               for other_column in range(self.num_columns )
               if other_column != column
           ]
           for other_row in range(self.num_rows )
           if other_row != row
       ]
       return Matrix(lowerCamelCase__ ).determinant()




    def      _a    (    self       ,						lowerCamelCase__       ,						lowerCamelCase__ ):
       if (row + column) % 2 == 0:
          return self.get_minor(lowerCamelCase__       ,						lowerCamelCase__ )
       return -1 * self.get_minor(lowerCamelCase__       ,						lowerCamelCase__ )




    def      _a    (    self ):
       return Matrix(
           [
               [self.get_minor(lowerCamelCase__       ,						lowerCamelCase__ ) for column in range(self.num_columns )]
               for row in range(self.num_rows )
           ] )




    def      _a    (    self ):
       return Matrix(
           [
               [
                   self.minors().rows[row][column]
                   if (row + column) % 2 == 0
                   else self.minors().rows[row][column] * -1
                   for column in range(self.minors().num_columns )
               ]
               for row in range(self.minors().num_rows )
           ] )




    def      _a    (    self ):
       lowerCAmelCase_:							Any       =	[
           [self.cofactors().rows[column][row] for column in range(self.num_columns )]
           for row in range(self.num_rows )
       ]
       return Matrix(lowerCamelCase__ )




    def      _a    (    self ):
       lowerCAmelCase_:							Dict       =	self.determinant()
       if not determinant:
          raise TypeError("Only matrices with a non-zero determinant have an inverse" )
       return self.adjugate() * (1 / determinant)




    def __repr__(    self ):
       return str(self.rows )




    def __str__(    self ):
       if self.num_rows == 0:
          return "[]"
       if self.num_rows == 1:
          return "[[" + ". ".join(str(self.rows[0] ) ) + "]]"
       return (
           "["
           + "\n ".join(
               [
                   "[" + ". ".join([str(lowerCamelCase__ ) for value in row] ) + ".]"
                   for row in self.rows
               ] )
           + "]"
       )




    def      _a    (    self       ,						lowerCamelCase__       ,						lowerCamelCase__ = None ):
       lowerCAmelCase_:							Tuple       =	TypeError("Row must be a list containing all ints and/or floats" )
       if not isinstance(lowerCamelCase__       ,						lowerCamelCase__ ):
          raise type_error
       for value in row:
          if not isinstance(lowerCamelCase__       ,						(int, float) ):
             raise type_error
       if len(lowerCamelCase__ ) != self.num_columns:
          raise ValueError(
              "Row must be equal in length to the other rows in the matrix" )
       if position is None:
          self.rows.append(lowerCamelCase__ )
       else:
          lowerCAmelCase_:							Tuple       =	self.rows[0:position] + [row] + self.rows[position:]




    def      _a    (    self       ,						lowerCamelCase__       ,						lowerCamelCase__ = None ):
       lowerCAmelCase_:							List[Any]       =	TypeError(
           "Column must be a list containing all ints and/or floats" )
       if not isinstance(lowerCamelCase__       ,						lowerCamelCase__ ):
          raise type_error
       for value in column:
          if not isinstance(lowerCamelCase__       ,						(int, float) ):
             raise type_error
       if len(lowerCamelCase__ ) != self.num_rows:
          raise ValueError(
              "Column must be equal in length to the other columns in the matrix" )
       if position is None:
          lowerCAmelCase_:							Union[str, Any]       =	[self.rows[i] + [column[i]] for i in range(self.num_rows )]
       else:
          lowerCAmelCase_:							Any       =	[
              self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
              for i in range(self.num_rows )
          ]




    def __eq__(    self       ,						lowerCamelCase__ ):
       if not isinstance(lowerCamelCase__       ,						lowerCamelCase__ ):
          return NotImplemented
       return self.rows == other.rows




    def __ne__(    self       ,						lowerCamelCase__ ):
       return not self == other




    def __neg__(    self ):
       return self * -1




    def __add__(    self       ,						lowerCamelCase__ ):
       if self.order != other.order:
          raise ValueError("Addition requires matrices of the same order" )
       return Matrix(
           [
               [self.rows[i][j] + other.rows[i][j] for j in range(self.num_columns )]
               for i in range(self.num_rows )
           ] )




    def __sub__(    self       ,						lowerCamelCase__ ):
       if self.order != other.order:
          raise ValueError("Subtraction requires matrices of the same order" )
       return Matrix(
           [
               [self.rows[i][j] - other.rows[i][j] for j in range(self.num_columns )]
               for i in range(self.num_rows )
           ] )




    def __mul__(    self       ,						lowerCamelCase__ ):
       if isinstance(lowerCamelCase__       ,						(int, float) ):
          return Matrix(
              [[int(element * other ) for element in row] for row in self.rows] )
       elif isinstance(lowerCamelCase__       ,						lowerCamelCase__ ):
          if self.num_columns != other.num_rows:
             raise ValueError(
                 "The number of columns in the first matrix must "
                 "be equal to the number of rows in the second" )
          return Matrix(
              [
                  [Matrix.dot_product(lowerCamelCase__       ,						lowerCamelCase__ ) for column in other.columns()]
                  for row in self.rows
              ] )
       else:
          raise TypeError(
              "A Matrix can only be multiplied by an int, float, or another matrix" )




    def __pow__(    self       ,						lowerCamelCase__ ):
       if not isinstance(lowerCamelCase__       ,						lowerCamelCase__ ):
          raise TypeError("A Matrix can only be raised to the power of an int" )
       if not self.is_square:
          raise ValueError("Only square matrices can be raised to a power" )
       if other == 0:
          return self.identity()
       if other < 0:
          if self.is_invertable():
             return self.inverse() ** (-other)
          raise ValueError(
              "Only invertable matrices can be raised to a negative power" )
       lowerCAmelCase_:							str       =	self
       for _ in range(other - 1 ):
          result *= self
       return result




    @classmethod
    def      _a    (    cls       ,						lowerCamelCase__       ,						lowerCamelCase__ ):
       return sum(row[i] * column[i] for i in range(len(lowerCamelCase__ ) ) )


if __name__ == "__main__":
 import doctest

 doctest.testmod()
def 		snake_case__	(		lowercase							,				lowercase					):

   if density <= 0:
      raise ValueError("Impossible fluid density"					)
   if bulk_modulus <= 0:
      raise ValueError("Impossible bulk modulus"					)

   return (bulk_modulus / density) ** 0.5


if __name__ == "__main__":
 import doctest

 doctest.testmod()