code: def ddm_berk(M, K):
      """
      Berkowitz algorithm for computing the characteristic polynomial.
  
      Explanation
      ===========
  
      The Berkowitz algorithm is a division-free algorithm for computing the
      characteristic polynomial of a matrix over any commutative ring using only
      arithmetic in the coefficient ring.
  
      Examples
      ========
  
      >>> from sympy import Matrix
      >>> from sympy.polys.matrices.dense import ddm_berk
      >>> from sympy.polys.domains import ZZ
      >>> M = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
      >>> ddm_berk(M, ZZ)
      [[1], [-5], [-2]]
      >>> Matrix(M).charpoly()
      PurePoly(lambda**2 - 5*lambda - 2, lambda, domain='ZZ')
  
      See Also
      ========
  
      sympy.polys.matrices.domainmatrix.DomainMatrix.charpoly
          The high-level interface to this function.
  
      References
      ==========
  
      .. [1] https://en.wikipedia.org/wiki/Samuelson%E2%80%93Berkowitz_algorithm
      """
      m = len(M)
      if not m:
          return [[K.one]]
      n = len(M[0])
  
      if m != n:
          raise DMShapeError("Not square")
  
      if n == 1:
          return [[K.one], [-M[0][0]]]
  
      a = M[0][0]
      R = [M[0][1:]]
      C = [[row[0]] for row in M[1:]]
      A = [row[1:] for row in M[1:]]
  
      q = ddm_berk(A, K)
  
      T = [[K.zero] * n for _ in range(n + 1)]
      for i in range(n):
          T[i][i] = K.one
          T[i + 1][i] = -a
      for i in range(2, n + 1):
          if i == 2:
              AnC = C
          else:
              C = AnC
              AnC = [[K.zero] for row in C]
              ddm_imatmul(AnC, A, C)
          RAnC = [[K.zero]]
          ddm_imatmul(RAnC, R, AnC)
          for j in range(0, n + 1 - i):
              T[i + j][j] = -RAnC[0][0]
  
      qout = [[K.zero] for _ in range(n + 1)]
      ddm_imatmul(qout, T, q)
      return qout
  
file_path: sympy/sympy/polys/matrices/dense.py
imports: from __future__ import annotations
  from operator import mul
  from .exceptions import DMShapeError, DMDomainError, DMNonInvertibleMatrixError, DMNonSquareMatrixError
  from typing import Sequence, TypeVar
  from sympy.polys.matrices._typing import RingElement
label_desc: Use image node_id 15 for calling a global function with example usage: ddm_berk(M, K) and returns: qout, list, list