1. Chapter 6 Matrices and Determinants Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 6.2 Inconsistent and Dependent Systems and Their Applications

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Apply Gaussian elimination to systems without unique solutions. Apply Gaussian elimination to systems with more variables than equations. Solve problems involving systems without unique solutions. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Inconsistent and Dependent Systems We can use Gaussian elimination on systems with three or more variables to determine how many solutions such systems may have. In the case of systems with no solution (inconsistent) or infinitely many solutions (dependent), it is impossible to rewrite the augmented matrix in the desired form with 1’s down the main diagonal from upper left to lower right, and 0’s below the 1’s.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: A System with No Solution Use Gaussian elimination to solve the system: Step 1 Write the augmented matrix for this system. Linear SystemAugmented Matrix

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: A System with No Solution (continued) Use Gaussian elimination to solve the system: Step 2 Attempt to simplify the matrix to row-echelon form, with 1s down the main diagonal and 0s below the 1s. replace row 2 by replace row 3 by

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: A System with No Solution (continued) Use Gaussian elimination to solve the system: Step 2 (cont) Attempt to simplify the matrix to row-echelon form, with 1s down the main diagonal and 0s below the 1s. replace row 3 by

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: A System with No Solution (continued) Use Gaussian elimination to solve the system: Step 2 (cont) Attempt to simplify the matrix to row-echelon form, with 1s down the main diagonal and 0s below the 1s. In equation form, the last row is This equation can never be a true statement. The solution set is the empty set,

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: A System with an Infinite Number of Solutions Use Gaussian elimination to solve the following system: We start with the augmented matrix replace row 2 byreplace row 3 by

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: A System with an Infinite Number of Solutions (continued) Use Gaussian elimination to solve the following system: replace row 2 byreplace row 3 by

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: A System with an Infinite Number of Solutions (continued) Use Gaussian elimination to solve the following system: The equation for row 3 is This equation results in a true statement regardless of which values are selected for x, y, and z. The equation is dependent on the other two equations in the system. The system can be expressed as the matrix: The result of Gaussian elimination on this system is the matrix:

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: A System with an Infinite Number of Solutions (continued) Use Gaussian elimination to solve the following system: The original system is equivalent to the system: The solution set is {11z + 13, 5z + 4, z)}.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: A System with Fewer Equations Than Variables Use Gaussian elimination to solve the system: We begin with the augmented matrix replace row 2 by This matrix translates into the system:

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: A System with Fewer Equations Than Variables (continued) Use Gaussian elimination to solve the system: The solution set for this system is {(z + 50, –2z + 10, z)}. The result of Gaussian elimination on this system is: