Applications

Involving Matrices…

Our last day in Sec. 7.2…

Right in with practice problems…:Happy Valley Farms produces three types of eggs: 1 (large),2 (X-large), 3 (jumbo). The number of dozens of type i eggssold to grocery store j is represented by a in the matrixij

100 60

A 120 70

200 120

The per dozen price Happy Valley Farms charges for egg type iis represented by b in the matrix

$0.80

B $0.85

$1.00

i 1

Find the product B AT

Right in with practice problems…the “Do Now”:

TB $0.80 $0.85 $1.00

What does this product represent???

T

100 60

B A $0.80 $0.85 $1.00 120 70

200 120

382 227.50

Each element in the product represents the total incomeEach element in the product represents the total incomeHappy Valley Farms makes at grocery store Happy Valley Farms makes at grocery store jj, selling all, selling allthree types of eggs.three types of eggs.

A company sells four models of one name brand “all-in-one fax,printer, copier, and scanner machine” at three retail stores. The inventory at store i of model j is represented by s in the matrix

ij

16 10 8 12

12 0 10 4

4 12 0 8

S

The wholesale and retail prices of model i are represented by p and p , respectively, in the matrix

$180 $269.99

$275 $399.99

$355 $499.99

$590 $799.99

P

i1 i2

Determine the product SP. What does this matrix represent?

16 10 8 12

12 0 10 4

4 12 0 8

S

$180 $269.99

$275 $399.99

$355 $499.99

$590 $799.99

P

$15,550 $21,919.54

$8,070 $11,439.74

$8,740 $12,279.76

SP

The wholesale and retail values of all the inventory at store i are represented by a and a , respectively, inthe matrix SP.

i1 i2

A building contractor has agreed to build six ranch-style houses,seven Cape Cod-style houses, and 14 colonial-style houses. Thenumber of units of raw materials that go into each type of houseare shown in the matrix

576

RanchR = Cape Cod

Colonial

222027

1410

8

795

172113

Steel Wood Glass Paint Labor

Assume that steel costs $1600 a unit, wood $900 a unit, glass$500 a unit, paint $100 a unit, and labor $1000 a unit.

1. Write a 1 x 3 matrix B that represents the number of each typeof house to be built.

6 7 14B

A building contractor has agreed to build six ranch-style houses,seven Cape Cod-style houses, and 14 colonial-style houses. Thenumber of units of raw materials that go into each type of houseare shown in the matrix

576

RanchR = Cape Cod

Colonial

222027

1410

8

795

172113

Steel Wood Glass Paint Labor

Assume that steel costs $1600 a unit, wood $900 a unit, glass$500 a unit, paint $100 a unit, and labor $1000 a unit.

2. Write a matrix product that gives the number of units of eachraw material needed to build the houses.

163 650 266 175 431BR

A building contractor has agreed to build six ranch-style houses,seven Cape Cod-style houses, and 14 colonial-style houses. Thenumber of units of raw materials that go into each type of houseare shown in the matrix

576

RanchR = Cape Cod

Colonial

222027

1410

8

795

172113

Steel Wood Glass Paint Labor

Assume that steel costs $1600 a unit, wood $900 a unit, glass$500 a unit, paint $100 a unit, and labor $1000 a unit.

3. Write a 5 x 1 matrix C the represents theper unit cost of each type of raw material.

$1600

$900

$500

$100

$1000

C

A building contractor has agreed to build six ranch-style houses,seven Cape Cod-style houses, and 14 colonial-style houses. Thenumber of units of raw materials that go into each type of houseare shown in the matrix

576

RanchR = Cape Cod

Colonial

222027

1410

8

795

172113

Steel Wood Glass Paint Labor

Assume that steel costs $1600 a unit, wood $900 a unit, glass$500 a unit, paint $100 a unit, and labor $1000 a unit.

4. Write a matrix product that gives the cost of each house.

$52,500

$56,100

$51,400

RC

A building contractor has agreed to build six ranch-style houses,seven Cape Cod-style houses, and 14 colonial-style houses. Thenumber of units of raw materials that go into each type of houseare shown in the matrix

576

RanchR = Cape Cod

Colonial

222027

1410

8

795

172113

Steel Wood Glass Paint Labor

Assume that steel costs $1600 a unit, wood $900 a unit, glass$500 a unit, paint $100 a unit, and labor $1000 a unit.

5. Compute the product BRC. What does this matrix represent?

$1,427,300BRC This is the building contractor’s totalThis is the building contractor’s total

cost of building all 27 houses.cost of building all 27 houses.

And now for a review of the properties of matrices:Let A, B, and C be matrices whose orders are such that thefollowing sums, differences, and products are defined.

1. Commutative Property

Addition:

A + B = B + A

Multiplication:

(Does not hold in general)

2. Associative Property

Addition:

(A + B) + C = A + (B + C)

Multiplication:

(AB)C = A(BC)

And now for a review of the properties of matrices:Let A, B, and C be matrices whose orders are such that thefollowing sums, differences, and products are defined.

3. Identity Property

Addition:

A + O = A

Multiplication: order of A = n x n

A I = I A = A

4. Inverse Property

Addition:

A + (–A) = O

Multiplication: order of A = n x n

AA = A A = I , |A| = 0

n n

n–1 –1

And now for a review of the properties of matrices:Let A, B, and C be matrices whose orders are such that thefollowing sums, differences, and products are defined.

5. Distributive Property

Multiplication over Addition

A(B + C) = AB + AC

(A + B)C = AC + BC

Multiplication over Subtraction

A(B – C) = AB – AC

(A – B)C = AC – BC