Systems of Equations

Matrix Methods

Introductions

What is a matrix?

A matrix is an organization of data into rows and columns

The size of a matrix is always listed

Rows x Columns

Place the following data into a matrix

2x + 3y = 6 -4x – 3y = 7

There are two ways to put this into matrix form.

We could use two matrices or put the information into one augmented matrix.

Augmented Matrix

A matrix which is used to represent systems of linear equations where a line is used to represent the location of the = sign. Coefficients are placed into the rows and columns of the matrix (coefficients from like variables must be in the same column).

Solving systems of equations using matrices

Two methods:– If write system into two matrix.

Solve by taking A-1B.

– If write system into one matrix. rref A– rref is reduced echelon form – this form puts 1’s

into the main diagonals and zeros in all other numerical positions

Examples of rref

Look at page 556 – top of page Look at page 557 – bottom of page

What happens if an entire row cancels out

Look at page 558 – middle of page

Solve: 2x + 3y – 6 = 0

4x – 6y + 2 =0

Solve

x – y + z = 10 3x + 3y = 5 x + y + 2z = 2

Adding and Subtracting MatricesScalars

Scalar is a constant which is multiplied times a matrix

To multiply by a scalar – multiply every term times the constant

To Add/Subtract Matrices Dimensions must be equal Add/subtract terms in same position

Example: Page 589 #11

Systems of Inequalities

Have to graph to solve– Graph each inequality on a coordinate plane:

Shade to appropriate area

– Solution is where the shaded areas overlap

Solve

2x – y < 4 3x + 2y > -6

Y > x + 2 Y < -x2 + 4

Applications: Linear Programming

A small farm in Illinois has 100 acres of land availiable on which to grow corn and soybeans. The following table shwos the cultivation cost per acre, the labor cost per acre, and the expected profit per acre. The column on the right shows the amount of money available for each of these expenses. Find the number of acres of each crop that should be planted to maximize profit.

Table

Soybeans Corn Money

Cultivation $40 $60 $1800

Labor $60 $60 $2400

Profit $200 $250

Recall that only 100 acres of land are available

Application 2

A meat market combines ground beef and ground port in a single package meat loaf. The ground beef is 75% lean and costs $.75 per pound. The ground pork is 60% lean and costs the market $.45 per pound. The meat loaf must be at least 70% lean. If the market wants to use at least 50 lb of its available pork, but no more than 200 lb of its beef, how much ground beef should be mixed with ground pork so the cost is minimized?

Application 3

An entrepreneur is having a design group produce at least six samples of a new kind of fastener that he wants to market. It costs $9.00 to produce each metal fastener and $4.00 to produce each plastic fastener. He wants to have at least two of each version of the fastener and needs to have all the samples 24 hours from now. It takes 4 hours to produce each metal sample and 2 hours to produce each plastic sample. To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost?

Assignment

Pg 472 #1-23 every 4th Pg 479

– #1-11 odd

Pg 590– #39-57 odd, 69, 71, 73

Pg 614– #23-39 every 4th, 43, 47, 51, 55

Pg 623– #21, 25, 29

Next Wednesday– Assignment notebook due– Chapter 5 and Matrix Test– Final

Many of the problems on final you have seen before. Study your old tests and look over your assignments.

Multiplying Matrices

To multiply matrices the inner dimensions must be equal

Ex: 2 x 3 times 3 x 4 Product will have the outer dimensions 2 x 4

Multiplying by Hand

Must multiply and entire row times an entire column to get one term

Go across first matrix and down second

Ex: Page 589 #15

Much easier to multiply on calculator

Always check dimensions – if dimensions do not work you can save yourself a lot of time.

Ex: Page 589 #14

Discriminant

Single value representing a matrix

2 x 2 determinant– Diagonal product down – diagonal product up

3 X 3 determinant– Minor method– Examples: Page 573: # 6, 13, 44, 53

Linear Programming

Process– Graph inequalities– Identify vertices (may have to use matrices to find

intersection points)– Place vertices into objective function– Identify maximum / minimum

– Examples: Page 622 #11, 13, 23, 27

Assignment

Page 544: #1,3,5,15,19,27,29,37,41,53,57,59,63,67,

71,75Page 561:

#1,9,13,29,33,45,49,55,63,75,79,81Page 573:

#5,13,47,51 Page 589: #7,17,27,59Page 622: 9,17,21,29