code: def _det_berkowitz(M):
      """Use the Berkowitz algorithm to compute the determinant."""
  
      if not M.is_square:
          raise NonSquareMatrixError()
  
      if M.rows == 0:
          return M.one
          # sympy/matrices/tests/test_matrices.py contains a test that
          # suggests that the determinant of a 0 x 0 matrix is one, by
          # convention.
  
      berk_vector = _berkowitz_vector(M)
      return (-1) ** (len(berk_vector) - 1) * berk_vector[-1]
  
imports: from types import FunctionType
  from sympy.core.cache import cacheit
  from sympy.core.numbers import Float, Integer
  from sympy.core.singleton import S
  from sympy.core.symbol import uniquely_named_symbol
  from sympy.core.mul import Mul
  from sympy.polys import PurePoly, cancel
  from sympy.functions.combinatorial.numbers import nC
  from sympy.polys.matrices.domainmatrix import DomainMatrix
  from sympy.polys.matrices.ddm import DDM
  from .exceptions import NonSquareMatrixError
  from .utilities import _get_intermediate_simp, _get_intermediate_simp_bool, _iszero, _is_zero_after_expand_mul, _dotprodsimp, _simplify
label_desc: Use image node_id 13 for calling a global function with example usage: _det_berkowitz(M) and returns: unknown, M