Generally speaking, when using Clem's rule to find solutions to a system of linear equations, the amount of calculation is relatively large. For specific numerical linear equations, computers can often be used to solve them when there are many unknowns. At present, there is a mature method for solving linear equations using computers.

　　Let a system of linear equations and its corresponding system of homogeneous linear equations be

 　　 (1)

  　　(2)

The determinant formed by the coefficient of a linear system of equations is called the coefficient determinant of the system of equations .

　　Clem's law gives the existence and uniqueness of a system of linear equations under certain conditions. Compared with its role in calculation, Clem's law has greater theoretical value. When judging the solution of a system of linear equations, based on four propositions and their relationships , Clem's law has the following commonly used expressions.

　　Theorem 1: If the coefficient determinant of the linear equation system (1) , then the linear equation system (1) must have a solution, and the solution is unique.

　　In problem solving or proof, the converse theorem of Theorem 1 is often used:

　　Theorem 2: If the linear equation system (1) has no solution or the solution is not unique, then its coefficient determinant must be zero.

　　Aligning the system of homogeneous linear equations (2), it is easy to see that must be the solution of this equation, which is called the zero solution of the system of homogeneous linear equations (2) . Applying Theorem 1 to the homogeneous linear equation system (2), the following conclusion can be obtained.
　　 Theorem 3: If the coefficient determinant of the homogeneous linear equation system (2) , then the homogeneous linear equation system (2) has only zero solution.
　　 Theorem 4: If the homogeneous linear equation system (2) has a non-zero solution, then its coefficient determinant .

　　Note: It will be further proved in the following chapters that if the coefficient determinant of the homogeneous linear equation system , then the homogeneous linear equation system (2) has a non-zero solution.