'''simple docstring'''






from __future__ import annotations




class  a			:



  """simple docstring"""







  def __init__(				self ,     snake_case_       ):

    '''simple docstring'''







    __UpperCAmelCase:       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:
      __UpperCAmelCase:       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
      __UpperCAmelCase:       List[str]		=			rows
    else:
      __UpperCAmelCase:       Optional[Any]		=			[]







  def  lowercase_	(				self       ):

    '''simple docstring'''







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







  @property
  def  lowercase_	(				self       ):

    '''simple docstring'''







    return len(self.rows       )







  @property
  def  lowercase_	(				self       ):

    '''simple docstring'''







    return len(self.rows[0]       )







  @property
  def  lowercase_	(				self       ):

    '''simple docstring'''







    return (self.num_rows, self.num_columns)







  @property
  def  lowercase_	(				self       ):

    '''simple docstring'''







    return self.order[0] == self.order[1]







  def  lowercase_	(				self       ):

    '''simple docstring'''







    __UpperCAmelCase:       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  lowercase_	(				self       ):

    '''simple docstring'''







    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  lowercase_	(				self       ):

    '''simple docstring'''







    return bool(self.determinant()       )







  def  lowercase_	(				self ,     snake_case_ ,     snake_case_       ):

    '''simple docstring'''







    __UpperCAmelCase:       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  lowercase_	(				self ,     snake_case_ ,     snake_case_       ):

    '''simple docstring'''







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







  def  lowercase_	(				self       ):

    '''simple docstring'''







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







  def  lowercase_	(				self       ):

    '''simple docstring'''







    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  lowercase_	(				self       ):

    '''simple docstring'''







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







  def  lowercase_	(				self       ):

    '''simple docstring'''







    __UpperCAmelCase:       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       ):

    '''simple docstring'''







    return str(self.rows       )







  def __str__(				self       ):

    '''simple docstring'''







    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  lowercase_	(				self ,     snake_case_ ,     snake_case_ = None       ):

    '''simple docstring'''







    __UpperCAmelCase:       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:
      __UpperCAmelCase:       Tuple		=			self.rows[0:position] + [row] + self.rows[position:]







  def  lowercase_	(				self ,     snake_case_ ,     snake_case_ = None       ):

    '''simple docstring'''







    __UpperCAmelCase:       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:
      __UpperCAmelCase:       Union[str, Any]		=			[self.rows[i] + [column[i]] for i in range(self.num_rows       )]
    else:
      __UpperCAmelCase:       Any		=			[
          self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
          for i in range(self.num_rows       )
      ]







  def __eq__(				self ,     snake_case_       ):

    '''simple docstring'''







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







  def __ne__(				self ,     snake_case_       ):

    '''simple docstring'''







    return not self == other







  def __neg__(				self       ):

    '''simple docstring'''







    return self * -1







  def __add__(				self ,     snake_case_       ):

    '''simple docstring'''







    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 ,     snake_case_       ):

    '''simple docstring'''







    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 ,     snake_case_       ):

    '''simple docstring'''







    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 ,     snake_case_       ):

    '''simple docstring'''







    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"""       )
    __UpperCAmelCase:       str		=			self
    for _ in range(other - 1       ):
      result *= self
    return result







  @classmethod
  def  lowercase_	(				cls ,     snake_case_ ,     snake_case_       ):

    '''simple docstring'''







    return sum(row[i] * column[i] for i in range(len(lowerCamelCase__       )       )       )


if __name__ == "__main__":
    import doctest

    doctest.testmod()
def 		snake_case__	(		lowercase					):
   lowerCAmelCase_:							Union[str, Any]       =	[1]

   lowerCAmelCase_  ,	lowerCAmelCase_  ,	lowerCAmelCase_:							int       =	0, 0, 0
   lowerCAmelCase_:							Union[str, Any]       =	ugly_nums[ia] * 2
   lowerCAmelCase_:							str       =	ugly_nums[ia] * 3
   lowerCAmelCase_:							Dict       =	ugly_nums[ia] * 5

   for _ in range(1							,				lowercase					):
      lowerCAmelCase_:							Any       =	min(lowercase							,				lowercase							,				lowercase					)
      ugly_nums.append(lowercase					)
      if next_num == next_a:
         ia += 1
         lowerCAmelCase_:							str       =	ugly_nums[ia] * 2
      if next_num == next_a:
         ia += 1
         lowerCAmelCase_:							Optional[int]       =	ugly_nums[ia] * 3
      if next_num == next_a:
         ia += 1
         lowerCAmelCase_:							int       =	ugly_nums[ia] * 5
   return ugly_nums[-1]


if __name__ == "__main__":
 from doctest import testmod

 testmod(verbose=True)
 print(f'''{ugly_numbers(2_0_0) = }''')