1. Copyright © 2011 Pearson, Inc. 7.3 Multivariate Linear Systems and Row Operations

2. Copyright © 2011 Pearson, Inc. Slide 7.3 - 2 What you’ll learn about Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.

3. Copyright © 2011 Pearson, Inc. Slide 7.3 - 3 Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system.

4. Copyright © 2011 Pearson, Inc. Slide 7.3 - 4 Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is 1. 3. The column subscript of the leading 1 entries increases as the row subscript increases.

5. Copyright © 2011 Pearson, Inc. Slide 7.3 - 5 Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.

6. Copyright © 2011 Pearson, Inc. Slide 7.3 - 6 Example Finding a Row Echelon Form

7. Copyright © 2011 Pearson, Inc. Slide 7.3 - 7 Example Finding a Row Echelon Form

8. Copyright © 2011 Pearson, Inc. Slide 7.3 - 8 Example Finding a Row Echelon Form

9. Copyright © 2011 Pearson, Inc. Slide 7.3 - 9 Example Finding a Row Echelon Form

10. Copyright © 2011 Pearson, Inc. Slide 7.3 - 10 Reduced Row Echelon Form If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

11. Copyright © 2011 Pearson, Inc. Slide 7.3 - 11 Invertible Square Linear System Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B, where X is the n  1 matrix of variables and B is the n  1 matrix of numbers on the right-hand side of the equations. If A –1 exists, then the system of equations has the unique solution X = A –1 B.

12. Copyright © 2011 Pearson, Inc. Slide 7.3 - 12 Example Solving a System Using Inverse Matrices

13. Copyright © 2011 Pearson, Inc. Slide 7.3 - 13 Example Solving a System Using Inverse Matrices

14. Copyright © 2011 Pearson, Inc. Slide 7.3 - 14 Example Fitting a Parabola to Three Points

15. Copyright © 2011 Pearson, Inc. Slide 7.3 - 15 Example Fitting a Parabola to Three Points

16. Copyright © 2011 Pearson, Inc. Slide 7.3 - 16 Example Fitting a Parabola to Three Points

17. Copyright © 2011 Pearson, Inc. Slide 7.3 - 17 Example Fitting a Parabola to Three Points

18. Copyright © 2011 Pearson, Inc. Slide 7.3 - 18 Quick Review

19. Copyright © 2011 Pearson, Inc. Slide 7.3 - 19 Quick Review Solutions