1. 2.2 Systems of Linear Equations: Unique Solutions

2. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Gauss-Jordan Elimination Method Operations 1. 1.Interchange any two equations. 2. 2.Replace an equation by a nonzero constant multiple of itself. 3. 3.Replace an equation by the sum of that equation and a constant multiple of any other equation.

3. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Solve the system Replace R 2 with [R 1 + R 2 ] Replace R 3 with [–2(R 1 ) + R 3 ] Replace R 2 with ½(R 2 ) 1 2 Row 1 (R 1 ) Row 2 (R 2 ) Row 3 (R 3 ) step...

4. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 5 3 Replace R 3 with [–3(R 2 ) + R 3 ] Replace R 3 with ½(R 3 ) Replace R 2 with [R 2 + R 3 ] Replace R 1 with [( –1) R 3 + R 1 ]...

5. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. So the solution is (3, –1, –2). 6 Replace R 1 with [R 2 + R 1 ]

6. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Row Operation Notation 1. 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. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Last example revisited: System Matrix...

9. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc....

10. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. This is in Row- Reduced Form

11. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Row–Reduced Form of a Matrix 1. 1.Each row consisting entirely of zeros lies below any other row with nonzero entries. 2. 2.The first nonzero entry in each row is a 1. 3. 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. 4.If a column contains a leading 1, then the other entries in that column are zeros.

12. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Row–Reduced Form of a Matrix Row-Reduced FormNon Row-Reduced Form R 2, R 3 switched Must be 0

13. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Gauss-Jordan Elimination Method 1. 1.Write the augmented matrix 2. 2.Interchange rows, if necessary, to obtain a nonzero first entry. Pivot on this entry. 3. 3.Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry. 4. 4.Continue until in row-reduced form.

16. Example Use the Gauss-Jordan elimination method to solve the system of equations Solution Example 5, page 82-83 The solution to the system is thus x = 3, y = 4, and z = 1.

17. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example A farmer has 200 acres of land suitable for cultivating crops A, B, and C. The cost per acre of cultivating crop A, crop B, and crop C is $40, $60, and $80, respectively. The farmer has $12,800 available for land cultivation. Each acre of crop A requires 20 labor-hours, each acre of crop B requires 25 labor-hours, and each acre of crop C requires 40 labor-hours. The farmer has a maximum of 6100 labor-hours available. If he wishes to use all of his cultivatable land, the entire budget, and all of labor available, how many acres of each crop should he plant?...

18. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution Let x = the number of acres of crop A Let y = the number of acres of crop B Let z = the number of cares of crop C Then we have: Use the Gauss-Jordan elimination method,... we have...

19. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution (cont.)... From the last augmented matrix in reduced form, we see that x = 50, y = 60, and z = 90. Therefore, the farmer should plant 50 acres of crop A, 60 acres of crop B, and 90 acres of crop C.