Download presentation

Presentation is loading. Please wait.

Published byGaven Comley Modified over 2 years ago

1
Section 2.2 Systems of Liner Equations: Unique Solutions

2
The Gauss-Jordan Elimination Method Operations 1.Interchange any two equations. 2.Replace an equation by a nonzero constant multiple of itself. 3.Replace an equation by the sum of that equation and a constant multiple of any other equation.

3
Ex. Solve the system Replace r2 with [r1 + r2] Replace r3 with [–2(r1) + r3] Replace r2 with ½(r2) 1 2 Row 1 (r1) Row 2 (r2) Row 3 (r3) step

4
4 5 3 Replace r3 with [–3(r2) + r3] Replace r3 with ½(r3) Replace r2 with [r2 + r3] Replace r1 with [( –1) r3 + r1]

5
So the solution is (3, –1, –2) 6 Replace r1 with [r2 + r1]

6
Augmented Matrix *Notice that the variables in the preceding example merely keep the coefficients in line. This can also be accomplished using a matrix. A matrix is a rectangular array of numbers System Augmented matrix coefficientsconstants

7
Row Operation Notation: 1.Interchange row i and row j 2.Replace row j with c times row j 3.Replace row i with the sum of row i and c times row j

8
Ex. Last example revisited: System Matrix

10
This is in Row- Reduced Form

11
Row–Reduced Form of a Matrix 1.Each row consisting entirely of zeros lies below any other row with nonzero entries. 2.The first nonzero entry in each row is a 1. 3.In any two consecutive (nonzero) rows, the leading 1 in the lower row is to the right of the leading 1 in the upper row. 4.If a column contains a leading 1, then the other entries in that column are zeros.

12
Row–Reduced Form of a Matrix Row-Reduced FormNon Row-Reduced Form R 2, R 3 switched Must be 0

13
Unit Column A column in a coefficient matrix where one of the entries is 1 and the other entries are 0. Unit columnsNot a Unit column

14
Pivoting – Using a coefficient to transform a column into a unit column This is called pivoting on the 1 and it is circled to signify it is the pivot.

15
Gauss-Jordan Elimination Method 1.Write the augmented matrix 2.Interchange rows, if necessary, to obtain a nonzero first entry. Pivot on this entry. 3.Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry. 4.Continue until in row-reduced form.

Similar presentations

Presentation is loading. Please wait....

OK

2.2 Systems of Linear Equations: Unique Solutions.

2.2 Systems of Linear Equations: Unique Solutions.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on tsunami in india 2004 Ppt on real estate marketing strategy Ppt on census and sample methods of collection of data Ppt on drainage system for class 9 Ppt on tribal communities of india Ppt on viruses and antivirus download Ppt on principles of object-oriented programming encapsulation Ppt on different solid figures games Ppt on regional trade agreements definition Ppt on air pollution act