1, when the coefficient determinant of the equation group is not equal to zero, the equation group has a solution and a unique solution;
2. If the equation has no solution or two different solutions, then the coefficient determinant of the equation must be equal to zero.
3. Cramer's Law is not only applicable to real number field, but also can be established in any field.
For systems with two or more equations, the calculation efficiency of Cramer's rule is very low; Compared with the elimination method of polynomial time complexity, its asymptotic complexity is O (n n! )。 Even for 2×2 systems, kramer's law is not stable numerically.
It is suitable for linear equations with the same number of variables and equations. It was published by the Swiss mathematician Clem (1704- 1752) in 1750' s Introduction to Linear Algebraic Analysis. In fact, Leibniz [1693] and Ma Kraulin [1748] also know this rule, but their notation is not as good as that of Clem.
Legal abstract of Kramer's law:
1, the important theoretical value of Cramer's rule: the relationship between the coefficient of the equation and the existence and uniqueness of the solution of the equation is studied; Compared with its role in calculation, Cramer's law has more important theoretical value.
2. Using Cramer's rule to judge the solutions of n equations and n unknown linear equations;
(1) When the coefficient determinant of the equation group is not equal to zero, the equation group has a solution and a unique solution;
(2) If the equations have no solution or two different solutions, then the coefficient determinant of the equations must be equal to zero.
(3) Cramer's law is not only applicable to the real number field, but also can be established in any field.
3. Limitations of Cramer's Law:
(1) When the number of equations of a system of equations is inconsistent with the number of unknowns, or when the determinant of the coefficient of the system of equations is equal to zero, the Cramer's law fails.
Effective.
(2) The amount of calculation is large, so it is necessary to calculate N+ 1 n-order determinants to solve an n-order linear equations.