Using Matrix Row Operations Maya and her friends Amit and Nina have a lawn care business offering three services: Lawn mowing and edging Fertilizing and weeding Trimming shrubs and small trees They charge a flat rate for each service. The three partners divide up the work for a particular customer as shown below Service Workers and Hours Cost Mowing Maya--1hr Amit--1hr Nina 1hr $21 Fertilizing Maya--2hr Amit--1hr $23 Amit-- 1hr Nina--3hr Trimming $25

Using Matrix Row Operations . Let m,a and n represent the hourly wages for Maya, Amit and Nina respectively. [ ] m+a+n=21 2m+a=23 A+3n=25 1 1 1 : 21 1 0 : 23 0 1 3 : 25 systems The final matrix is said to be in reduced row-echelon form. To transform an augmented matrix into reduced row-echelon Form, use the elementary row operations.

Using Row Matrix Operations Elementary Row Operations The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form. Interchange two rows Multiply all entries in one row by a non-zero number. Add a multiple of one row to another.

Using Matrix Row Operations Refer to the lawn-care problem described at the beginning of the lesson. Use the row-reduction method to solve the system. Find the hourly wages for Maya, Amit, and Nina. Then find the total amount that each partner earns for this job. [ ] m+a+n=21 2m+a=23 A+3n=25 1 1 1 : 21 1 0 : 23 0 1 3 : 25 systems Perform row operations Inspect column 1

Using Matrix Row Operations The first row begins with a 1, but the 2 in the second row needs to become a 0. Identity Solutions [ ] 1 1 1 : 21 1 0 : 23 0 1 3 : 25 [ ] 1 0 0 : 8 0 1 0 : 7 0 0 1 : 6 We want to replace row 2with The sum of row 2 and -2 times row 1 This is our goal we want our Matrix to look like [ ] 1 1 1 : 21 0 -1 -2 : -19 0 1 3 : 25 -2R1+R2 R2

Using Matrix Row Operations [ ] Inspect column 2 Row 1 Change it to zero 1 1 1 : 21 0 -1 -2 : -19 0 1 3 : 25 R2 + R1 R1 [ ] 1 0 -1 : 2 0 -1 -2 : -19 0 1 3 : 25

Using Matrix Row Operations [ ] Inspect column 2 Row 2 Change it to zero 1 0 -1 : 2 0 -1 -2 : -19 0 1 3 : 25 -1R2 R2 [ ] 1 0 -1 : 2 0 1 2 : 19 0 1 3 : 25

Using Matrix Row Operations [ ] Inspect column 2 Row 3 Change it to zero 1 0 -1 : 2 0 1 2 : 19 0 1 3 : 25 -1R2+R3 R2 [ ] 1 0 -1 : 2 0 1 2 : 19 0 0 1 : 6

Using Matrix Row Operations [ ] Inspect column 3 Row 1 Change it to zero 1 0 -1 : 2 0 1 2 : 19 0 0 1 : 6 R3+R1 R1 [ ] 1 0 0 : 8 0 1 2 : 19 0 0 1 : 6

Using Matrix Row Operations [ ] Inspect column 3 Row 3 Change it to zero 1 0 0 : 2 0 1 2 : 19 0 0 1 : 6 -2R3+R2 R2 [ ] 1 0 0 : 8 0 1 0 : 7 0 0 1 : 6 The matrix is now row echelon form Maya=$8hr Amit=$7hr Nina=$6