2. Gauss-Jordan Matrix Elimination Brought to you by Tutorial Services – The Math Center

3. Gauss-Jordan Matrix Elimination A method that can be used to solve systems of linear equations involving two or more variables. To do so, the system must be changed first, to an augmented matrix.

4. Augmented Matrix a 1 x +b 1 y +c 1 z =d 1 a 2 x +b 2 y +c 2 z =d 2 a 3 x +b 3 y +c 3 z =d 3 System of Equations  Augmented Matrix 

5. Example System of Equations  Augmented Matrix 

6. Elementary Row Operations 1.Interchanging two rows. 2.Adding one row to another row, or multiplying one row by a constant first and then adding it to another. 3.Multiplying a row by any constant different from zero.

7. Gauss-Jordan Matrix Elimination Goal In order to solve the system of equations, a series of steps needs to be followed using the elementary row operations. The reduced matrix should end up being the identity matrix.

8. Identity Matrix Identity Matrix for a 3 x 3 Identity Matrix for a 4 x 4

9. Solving the System 1. Write as an augmented Matrix2. Switch row 1 with row 2

10. 3. Multiply Row 1 by -3 and add Row 2 R 1 (-3) -33-6-12 + 32-1 3 05-7 -9 R2R2 R2R2 R 1 (-3) + R 2  R 2

11. 4. Multiply Row 1 by -2 and add Row 3 R 1 (-2) -22-4-8 + 23-1 3 05-5 -5 R3R3 R3R3 R 1 (-2) + R 3  R 3

12. 5. Switch Row 2 with Row 3 6. Multiply Row 2 by 1/5 R 2 (1/5 )  R 2 R 2 R 3

13. 7. Add Row 2 to Row 1 R 1 + R 2  R 1 8. Multiply Row 2 by -5 and Add Row 3 R 2 (-5) + R 3  R 3

14. 10. Add Row 3 and Row 2 9. Multiply Row 3 by -1/2 R 3 ( -1/2 )  R 3 R 3 + R 2  R 2

15. Final Answer 11. Multiply Row 3 by -1 and add Row 1 R 3 (-1) + R 1  R 1

16. Gauss Jordan Handouts and Links Gauss Jordan Method Handout Adding and Subtracting Matrices Workshop Adding and Subtracting Matrices Handout Multiplying Matrices Workshop Multiplying Matrices Handout Inverse Matrix Handout