Multivariate Linear Systems and Row Operations

1. Precalculus Lesson 7.3 Multivariate Linear Systems and Row Operations

2. Quick Review

3. 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.

4. Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. • Interchange any two equations of the system. • Multiply (or divide) one of the equations by any nonzero real number. • Add a multiple of one equation to any other equation in the system. Transforming a system to triangular form is called Gaussian elimination.

5. Using Gaussian Elimination Solve the system of equations using Gaussian elimination. Mult. 1st equation by -2 and add to 2nd equation, replacing the 2nd. Mult. 1st equation by -4 and add to 3rd equation, replacing the 3rd. Mult. 2nd equation by -1 and add to 3rd equation, replacing the 3rd. Triangular form makes the solution easy to read.

6. Using Gaussian Elimination Solve the system of equations using Gaussian elimination. Mult. 1st equation by -2 and add to 2nd equation, replacing the 2nd. Mult. 1st equation by 3 and add to 3rd equation, replacing the 3rd. Mult. 2nd equation by 2 and add to 3rd equation, replacing the 3rd.

7. Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. • Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. • The first entry in any row with nonzero entries is 1. • The column subscript of the leading 1 entries increases as the row subscript increases. Another way to phrase parts 2 and 3 is to say that the leading 1’s move to the right as we move down the rows.

8. Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. • Interchange any two rows. • Multiply all elements of a row by a nonzero real number. • Add a multiple of one row to any other row.

9. Example Finding a Row Echelon Form The augmented matrix of this system of equations is:

10. Example Finding a Row Echelon Form Indicates interchanging the ith and jth row of the matrix. Indicates multiplying the ith row by −2.

11. Example Finding a Row Echelon Form Indicates multiplying the ith row by −3 and adding it to the jth row.

12. Example Finding a Row Echelon Form

13. Example Finding a Row Echelon Form

14. 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.

15. Example Reduced Row Echelon Form

16. Example Reduced Row Echelon Form

17. Example Reduced Row Echelon Form

18. Homework: Text pg602 Exercises #4, 6, 10, 12, 18