What is the formula of Cramer's Law?

Cramer's rule, also translated as Cramer's rule, is a theorem about solving linear equations in linear algebra. It is suitable for linear equations with the same number of variables and equations. It was published by the Swiss mathematician Clem in 1750's Introduction to Linear Algebraic Analysis. In fact, Leibniz and Ma Kraulin also know this rule, but their notation is not as good as Clem's.

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.

Related information:

Generally speaking, when solving linear equations by Cramer's law, the calculation amount is relatively large. The time complexity of the algorithm for finding the solution of linear equations by Cramer's rule depends on the algorithm complexity O(f(n)) of matrix determinant, which is o (n f (n)), which is generally of no computational value and too complicated.

For specific numerical linear equations, when there are many unknowns, it can often be solved by computer. At present, there is a set of mature methods to solve linear equations by computer.