1. F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Inverse of Matrix. Gauss-Jordan Elimination Part 4

2. How do we find the inverse for 3 x 3 ??? The Inverse of a Matrix 3X3 (No Calculator)

3. The 3 x 3 Inverse Note : there are two methods explained in your text book. We will start with the one on Page 317 “ Determination of the Inverse by Gauss-Jordan Method ”

4. Determination of the Inverse : Gauss-Jordan Elimination AX = I I X = K I X = X = A -1 => K = A -1 1) Augmented matrix all A, X and I are (nxn) square matrices X = A -1 Gauss eliminationGauss-Jordan elimination UT: upper triangular further row operations [A I ] [ UT H] [ I K] 2) Transform augmented matrix Wilhelm Jordan (1842– 1899)

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

6. 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)

7. Use the Gaussian-Jordon elimination to find the inverse matrix Ex ： Find the inverse of Sol: 12

8. 1 2 Hence

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

10. Ex ： Find the inverse matrix of Sol: Using Gauss-Jordan Elimination

12. Thus

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

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

15. Find A -1 using the Gauss-Jordan method. Example Process: Expand A|I. Start scaling and adding rows to get I|A -1.

16. Gauss-Jordan Elimination

18. Example Find the inverse of Step 1: Step 2:

19. Example Inverses (2) Step 3: