code: def test_DDM_lu_solve():
      # Basic example
      A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
      b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
      x = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
      assert A.lu_solve(b) == x
  
      # Example with swaps
      A = DDM([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
      assert A.lu_solve(b) == x
  
      # Overdetermined, consistent
      A = DDM(
          [[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ
      )
      b = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
      assert A.lu_solve(b) == x
  
      # Overdetermined, inconsistent
      b = DDM([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
      raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
  
      # Square, noninvertible
      A = DDM([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
      b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
      raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
  
      # Underdetermined
      A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
      b = DDM([[QQ(3)]], (1, 1), QQ)
      raises(NotImplementedError, lambda: A.lu_solve(b))
  
      # Domain mismatch
      bz = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
      raises(DMDomainError, lambda: A.lu_solve(bz))
  
      # Shape mismatch
      b3 = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
      raises(DMShapeError, lambda: A.lu_solve(b3))
  
imports: from sympy.testing.pytest import raises
  from sympy.external.gmpy import GROUND_TYPES
  from sympy.polys import ZZ, QQ
  from sympy.polys.matrices.ddm import DDM
  from sympy.polys.matrices.exceptions import DMShapeError, DMNonInvertibleMatrixError, DMDomainError, DMBadInputError
label_desc: Use image node_id 25 for calling a global function with example usage: test_DDM_lu_solve() without return types