Inverse and Elementary Matrices

1. Inverse and Elementary Matrices

2. 反矩陣 (inverse matrix) Let If there is a matrix such that then (1) A is called an invertible or nonsingular matrix (2) B is the inverse matrix of A • Note: If a matrix having no inverse matrix is called a noninvertible or singular matrix.

3. Theorem： If B and C are both the inverse of A, then B = C Pf: Since B = C, the inverse matrix of a matrix is unique. • Note: (1) The inverse matrix of A is denoted by (2)

4. Use the Gaussian-Jordon elimination to find the inverse matrix • Ex：Find the inverse of Sol: 1 2

5. 1 2 Hence

6. Note： If A can’t using row operations to be translated into identity matrix I, then A is a singular matrix.

7. Ex：Find the inverse matrix of Sol:

9. Thus

10. The power of a square matrix

11. Theorem： If A is invertible, then the following properties hold:

12. Theorem： • If A and B are both invertible with the size nn, then • AB is invertible and Pf: Thus AB is invertible and the inverse matrix of AB is (BA)1. • Note:

13. Theorem：Cancellation laws • If C is invertible, then the following properties hold: • (1) If AC=BC, then A=B • (2) If CA=CB, then A=B Pf: Since C is invertible, C1 exists. • Note: If C is noninvertible, then the cancellation laws do not hold.

14. Theorem： If A is invertible, then Ax = b has a unique solution . Pf: ( A is nonsingular) If x1 and x2 are two solutions of Ax = b, then Ax1 = b = Ax2. By the cancellation law, x1 = x2, the solution is unique. • Note:

15. row elementary matrix(列基本矩陣) • An nn matrix is called a row elementary matrix if it can be attained from identity I by doing only one row elementary operation • Three different row elementary matrices identity matrix elementary matrix Elementary operation

16. Ex： • (a) (c) (b) (d) (e) (f)

17. Ex：求一序列的基本矩陣以將下列矩陣化簡成列梯形形式Ex：求一序列的基本矩陣以將下列矩陣化簡成列梯形形式 Sol:

18. = B

19. 列等價 (row-equivalent) If there are finite row elementary matrices, E1, E2, …, Ek such that , then B is row-equivalent to A.

20. Note:

21. Ex: • elementary matrixinverse matrix

22. Assume that A can be written as a product of a sequence • of elementary matrices. Every elementary matrix is • invertible. The product of invertible matrices is invertible. • Thus A is invertible. Pf: (2) If A is invertible then Ax = 0 has only trivial solution. Thus A can be written as a product of elementary matrices.

23. Ex：Find a sequence of elementary matrices such that their product is the given matrix Sol:

24. Theorem: If A is an nn matrix, then the following statements are equivalent: (1) A is invertible. (2) For any n1 matrixb, Ax = b has only one solution. (3) Ax = 0 has only trivial solution. (4) A is (row) equivalent to In. (5) A can be represented as a product of elementary matrices.

25. LU-factorization (LU-分解) Represent an nn matrixA as a product of a lower triangular matrix L and an upper triangular matrix U . L is a lower triangular matrix U is an upper triangular matrix • Note: If A is LU-factorizatiable than we can use only one elementary operation, rij(k), to translate A into LU.

26. Ex: Do the LU-factorization of A. (a) (b) Sol: (a)

27. (b)

28. Use LU-factorization to solve Ax=b. • Two steps: (1) Let y=Ux. Solve Ly=b, find y. (2) Solve Ux=y, then we can get x.

29. Ex： Use LU-factorization to solve the given linear system. Sol: (1) Let y=Ux. Solve Ly=b, find y.

30. (2) Solve Ux=y, then we can get x. Thus the solution of the given system is