1. Lecture 6 Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng

2. Expressing Elimination by Matrix Multiplication

3. Elementary or Elimination Matrix The elementary or elimination matrix That subtracts a multiple l of row j from row i can be obtained from the identity matrix I by adding (-l) in the i,j position

4. Elementary or Elimination Matrix

5. Pivot 1: The elimination of column 1 Elimination matrix

6. The Product of Elimination Matrices

7. Elimination by Matrix Multiplication

8. Linear Systems in Higher Dimensions

11. Booking with Elimination Matrices

12. Multiplying Elimination Matrices

13. Inverse Matrices In 1 dimension

14. Inverse Matrices In high dimensions

15. Inverse Matrices In 1 dimension In higher dimensions

16. Some Special Matrices and Their Inverses

17. Inverses in Two Dimensions Proof:

18. Uniqueness of Inverse Matrices

19. Inverse and Linear System

20. Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed)

21. Inverse, Singular Matrix and Degeneracy Suppose there is a nonzero vector x such that Ax = 0 [column vectors of A co-linear] then A cannot have an inverse Contradiction: So if A is invertible, then Ax =0 can only have the zero solution x=0

22. One More Property Proof So

23. Gauss-Jordan Elimination for Computing A -1 1D 2D

24. Gauss-Jordan Elimination for Computing A -1 3D

25. Gauss-Jordan Elimination for Computing A -1 3D: Solving three linear equations defined by A simultaneously n dimensions: Solving n linear equations defined by A simultaneously

26. Example:Gauss-Jordan Elimination for Computing A -1 Make a Big Augmented Matrix

27. Example:Gauss-Jordan Elimination for Computing A -1