Other equivalent conditions of matrix invertibility;
1, the column (row) vector group of a square matrix A is linearly independent, that is to say, the Ax=0 equation group has only zero solution.
2. According to Cramer's law, if the homogeneous linear equations have only zero solutions, the coefficient determinant is not zero.
3. If the determinant is not zero, it is a necessary and sufficient condition for the matrix to be invertible.
To sum up, if the row vector group and column vector group of A are linearly independent, then the matrix A is reversible.
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 (1704- 1752) in 1750' s Introduction to Linear Algebraic Analysis.
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.
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.
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.
(2): The amount of calculation is large, so to solve an n-order linear equations, it is necessary to calculate N+ 1 n-order determinants.