1. adjoint matrix Yes! The array of algebraic complements!

2. example: Write the adjoint matrix of a 2*2 matrix.

3. An important formula!

5. 1.The elements are arranged as ascending order, and exponentials are arithmetical series. 2.The result may be positive,negative or zero. 3. A product of n(n-1)/2 items. Our task is to calculate other determinants by Vedermonde determinant. So we should memorize the form and result of Vedermonde determinant. Can you identify Vedermonde determinant ？ Can you solve problems by the result of Vedermonde determinant ？

6. Such as ：

8. Cramer principle Consider the following linear system: Similar to binary system, the nth element equation can be expressed by determinant.

9. Theorem 1. If the coefficient determinant then the system has unique solution:

10. where There are two propositions to prove. One is the existence of solution, the other is the uniqueness. And the solution can be written as

11. To prove are solutions, we should prove the equations From the equation above, we can get Therefore we construct a (n+1)th order determinant

12. This determinant equals zero. Compute it by the 1st row, we obtain

14. Then prove the uniqueness of solution, and Bythe first proposition is proved. that is

16. Theorem 2. If the coefficient determinant is nonzero, then the system has only one solution. System: is called homogeneous linear system.

17. Theorem 3. If the coefficient determinant of a homogeneous linear equation then the system has one unique solution.

18. So when the system has nonzero solution. example 2 ： Prove that the following system has zero solution only. Solve: If the system has nonzero solution, the coefficient determinant must be zero.

21. So the system has zero solution only. Practice determinant ： 1. The (n+1)th column plus to the nth,  the 2nth column plus to the 1st one.

22. The question can also be solved by computing it according to the row or column. Compute the determinant by the 1st row.

24. (compute it by the 1 st row)

25. Keep the method in heart!