Larger Systems of Linear Equations and Matrices Section 6.2 Larger Systems of Linear Equations and Matrices --- Gauss-Jordan Method Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix. A second method of elimination, called Gauss-Jordan elimination, continues the reduction process until a reduced row-echelon form is obtained. This procedure is demonstrated in Example 8.
Example– Gauss-Jordan Elimination Use Gauss-Jordan elimination to solve the system x – 2y + 3z = 9 –x + 3y = –4 2x – 5y + 5z = 17 Solution: The row-echelon form of the linear system can be obtained using the Gaussian elimination. .
Example – Solution cont’d Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows. Perform operations on R1 so second column has a zero above its leading 1. Perform operations on R1 and R2 so third column has zeros above its leading 1.
Example – Solution The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, you have x = 1 y = –1. z = 2 Now you can simply read the solution, x = 1, y = –1, and z = 2 which can be written as the ordered triple(1, –1, 2). cont’d
Reduced Row Echelon Form : Gauss-Jordan Method Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
The system is inconsistent. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.