2.2 Systems of Linear Equations: Unique Solutions.

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2.2 Systems of Linear Equations: Unique Solutions

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

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...

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

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 ]

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

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

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

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

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

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 The first nonzero entry in each row is a 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 If a column contains a leading 1, then the other entries in that column are zeros.

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

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

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.

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 Interchange rows, if necessary, to obtain a nonzero second entry in the second row. Pivot on this entry Continue until in row-reduced form.

Example Use the Gauss-Jordan elimination method to solve the system of equations Solution Example 5, page 82-83

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?...