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How to understand the relationship between Cramer's law coefficient and solution

Cramer's rule studies the relationship between the coefficient of an equation and the existence and uniqueness of its solution.

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. When there is no solution, the equations are said to be incompatible or inconsistent, and when there are multiple solutions, it is called uncertainty. For linear equations, an uncertain system will have infinite solutions (if in infinite domain), because the solutions can be expressed by one or more parameters that can take any value.

4. Cramer's Law is not only applicable to real number field, but also can be established in any field.

Extended data

Limitations of Cramer's Law:

Kramer's law applies to the case that the determinant of coefficient is not zero. In the case of 2×2, the coefficient determinant is zero, the molecular determinant is non-zero and the system is incompatible.

(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.

(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.

Baidu Encyclopedia-Cramer Rule