Deck 7: Matrix Algebra

Question 1

Find the transpose of the matrix:

Answer 1

Question 2

Write a diagonal matrix of size 3 × 3 with entries
= i + 3,
= 0 for i ≠ j.

Answer 2

Question 3

Find transpose of
.

Answer 3

Question 4

Solve the matrix equation:
=

Question 5

Write A =
if A is 2 × 2 and
= 2i + j.

Question 6

A manufacturer of blended juices will also bottle pure juices for one supermarket chain.This week he bottles cherry,grape,and cranberry juice.Fill in the matrix for the gallons of each juice needed to produce 1 gallon of juice for each of the 3 pure juices.

Question 7

If A =
,determine (a)
,and (b)the order of A.

Question 8

If =
,then y =

A) 0.
B) 2.
C) -2.
D) 4.
E) -4.

Question 9

A manufacturer who produces products A,B,and C is only interested in tracking labor costs.What is the order of the matrix he would use?

Question 10

A manufacturer who requires raw materials A,B,C,D,and E is interested in tracking the costs of these materials from 3 different sources.What is the order of the matrix he would use?

Question 11

A manufacturer who produces one product is only interested in tracking material costs.What is the order of the matrix he would use?

Question 12

If A =
,determine (a)
,and (b)the order of A.

Question 13

Find a 3 × 2 matrix
with
=
= 0,all other
equal to 1.

Question 14

Find the transpose of
.

Question 15

Use the matrix operations on a graphing calculator to find the transpose of

Question 16

Solve the matrix equation:
=

Question 17

A blended juice product requires 5 gallons of apple juice,no orange juice,and 2 gallons of cranberry juice.Construct an ingredient matrix for this product.

Question 18

An employer offers medical,dental,and life insurance to both salaried and unsalaried employees.She describes her annual costs for these benefits in a matrix,and finds that,after employee contributions,only medical insurance requires an employer contribution of $20,000 for salaried and $15,000 for unsalaried employees.Construct a matrix which shows this result.

Question 19

Write A =
if A is 2 × 3 and
= 2i + j.

Question 20

If A = where A is 2 × 2 and
= i - 2j,then matrix A is given by

A)
B)
C)
D)
E)

Question 21

Solve the matrix equation: 3
- 2
=

Question 22

Perform the indicated operations and simplify your answer: 3
- 4

Question 23

An office furniture company manufactures desks and tables at two plants,A and B.Matrix J represents the production of the two plants in January,and matrix F represents the production of the two plants in February.Use the matrix operations on a graphing calculator to write a matrix that represents the total product at the two plants for the two months.
J =
; F =

Question 24

Show that A + O = A.

Question 25

Perform the indicated operations and simplify your answer:
+ 2

Question 26

If 4 + 2
=
,then

A) x = 0 and y = 0.
B) x = 4 and y = -5.
C) x = 0 and y = 4.
D) x = and y =
E) There are no values for x and y which satisfy the equation.

Question 27

A pharmaceutical company manufactures 3 drugs for high blood pressure,A,B,and C,each of which requires a 2-step production process.The production cost of each drug has a materials and labor component.If the matrices
and
represent the costs per ounce associated with each step,use the matrix operations on a graphing calculator to write a matrix that shows the total cost of the two steps.

=
;
=

Question 28

What is the total demand for the four types of energy from only the energy industries?

Question 29

A small airline has 3 flights,A,B,and C,to each of 3 cities in Colorado.The matrix N represents the number of passengers carried in November,and matrix D represents the number of passengers carried in December.Write a matrix that shows the total number of passengers carried in these two months.
N =
; D =

Question 30

Consider the matrix ,this matrix can be best described as a(n):

A) upper triangular matrix
B) lower triangular matrix
C) diagonal matrix
D) main diagonal

Question 31

Use the matrix operations on a graphing calculator to find the total demand for the four types of energy from only the consumers.

Question 32

Perform the indicated operations and simplify your answer: 2

Question 33

Use the matrix operations on a graphing calculator to show that A + (B +
B) + C.
C) = (A +

Question 34

If A =
; B =
,then find (A +
.

Question 35

Show that A + B = B + A.

Question 36

Use the matrix operations on a graphing calculator to find the total demand for the four types of energy from the manufacturing facility and the university.

Question 37

Use the matrix operations on a graphing calculator to show that A + C = C + A.

Question 38

Find x,y,z,u,t,v such that
+ 2
=

Question 39

Solve the matrix equation: x
+ 2
+ y
=

Question 40

2 -
=

A)
B)
C)
D)
E) none of the above

Question 41

If A =
and B =
,find A + B.

Question 42

What is the change in sales between 1994 and 1995?

Question 43

Perform the indicated operation and simplify your answer:

Question 44

The researcher then changes the original demand matrix using the matrix equation
= A + 0.2D,where A =
.Use the matrix operations on a graphing calculator to calculate the new demand matrix in his model.

Question 45

If A =
and B =
,find A - B.

Question 46

Use the matrix operations on a graphing calculator to find her income if increasing it by 20% would cause an increase in profit of 50%.(Round to the nearest thousand dollars.)

Question 47

A new EPA mandate requires that all energy users reduce their energy use from the original level by
over the next 10 years to minimize global warming.Use the matrix operations on a graphing calculator to calculate a new demand matrix which reflects this decrease.

Question 48

If A =
,B =
,and C =
,find 7A + 6B - 3C.

Question 49

What is the change in sales between 1994 and 1996?

Question 50

Use the matrix operations on a graphing calculator to find her income if increasing it two times would cause her profit to be three times as large.(Round to the nearest thousand dollars.)

Question 51

What is the total demand for the four types of energy from both the consumers and the energy industries?

Question 52

A boom in the economy causes an across the board increase of 10% in energy demands in the area.Calculate a new demand matrix which reflects this increase.

Question 53

A researcher is studying the effects on the local economy of changes in the original demand matrix.He models one change by the matrix equation
= D + 0.5D.Calculate the new demand matrix in his model.

Question 54

Perform the indicated operation and simplify your answer: [6][5]

Question 55

If A =
and B =
,find -4A + 5B.

Question 56

What would her income be if increasing it threefold would cause her profit to be five times as large? (Round to the nearest thousand dollars.)

Question 57

If A =
,B =
,and C =
,find 4A - 2B + 3C.

Question 58

What is the change in sales between 1995 and 1996?

Question 59

If A =
and B =
,find 2A - 3B.

Question 60

Perform the indicated operation and simplify your answer:

Question 61

Three components A,B,and C,are used in the manufacturing of a product.For a production run,100,150,and 200 units of A,B,and C are $2,$1,and $3,respectively.Represent the total cost of A,B,and C by a matrix product.

Question 62

Perform the indicated operation if possible:

Question 63

=

A)
B)
C)
D)
E)

Question 64

A bookstore has 100 dictionaries,70 cookbooks,and 90 thesauruses in stock,and a second store with 85 dictionaries,95 cookbooks,and 60 thesauruses in stock.If the value of each dictionary is $28,each cookbook is $22 and each thesaurus is $16,use the matrix operations on a graphing calculator to find the total value of the bookstore's inventory.

Question 65

An electronics store has 35 televisions,15 VCRs,and 25 CD players in stock,and a second store with 45 televisions,25 VCRs,and 30 CD players in stock.If the value of each television is $400,each VCR is $200 and each CD player is $150,find the total value of the electronics store's inventory.

Question 66

The prices (in dollars per unit)for 3 video tapes are represented by the price vector:
A video rental store orders these tapes in the quantities given by the column vector:
Find the total cost (in dollars)of the purchase.

Question 67

=

A)
B)
C)
D)
E) none of the above

Question 68

Let A =
.Find
.

Question 69

Use matrix multiplication to represent the system:

Question 70

Give an example of 2 matrices A and B where A × B ≠ B × A.

Question 71

If A is a 3 × 5 matrix,B is a 5 × 7 matrix,C is a 7 × 11 matrix,D is a 5 × 7 matrix,which of the following matrix products are defined? Give the size of each answer if it is defined.
(a)A × B × C
(b)B × C × D
(c)A × D × C
(d)A × B × D

Question 72

A pet store has 6 kittens,10 puppies,and 7 parrots in stock,and a second store with 8 kittens,14 puppies,and 9 parrots in stock.If the value of each kitten is $55,each puppy is $150 and each parrot is $35,find the total value of the pet store's inventory.

Question 73

An appliance store has 25 refrigerators,30 ranges,and 10 dishwashers in stock.If the value of each refrigerator is $600,each range is $300 and each dishwasher is $250,use matrix operations to find the total value of the appliance store's inventory.

Question 74

Perform the indicated operations and simplify your answer: 2
-

Question 75

An electronics store has 35 televisions,15 VCRs,and 25 CD players in stock.If the value of each television is $400,each VCR is $200 and each CD player is $150,use the matrix operations on a graphing calculator to find the total value of the electronics store's inventory.

Question 76

The prices (in dollars per unit)for 3 textbooks are represented by the price vector:
P =
.A university bookstore orders these books in the quantities given by the column vector: Q =
.Use the matrix operations on a graphing calculator to find the total cost (in dollars)of the purchase.

Question 77

If the orders of the matrices Q,R,and S are 4 × 5,3 × 2,and 5 × 3,respectively,then the order of QRS is

A) 4 × 5.
B) 3 × 5.
C) 3 × 3.
D) 2 × 4.
E) none of the above

Question 78

An appliance store has 25 refrigerators,30 ranges,and 10 dishwashers in stock,and a second store with 15 refrigerators,25 ranges,and 20 dishwashers in stock.If the value of each refrigerator is $600,each range is $300 and each dishwasher is $250,find the total value of the inventory at the two appliance stores.

Question 79

Let A =
; B =
; C =
.Find A

Question 80

Use matrix multiplication to represent the system: