Finite Mathematics and Applied Calculus

Stefan Waner, Steven Costenoble

Chapter 3

Matrix Algebra and Applications - all with Video Answers

Educators

Section 1

Matrix Addition and Scalar Multiplication

Problem 1

Find the dimensions of the given matrix and identify the given entry.
$$
A=\left[\begin{array}{llll}
1 & 5 & 0 & \frac{1}{4}
\end{array}\right] ; a_{13}
$$

Laurie Huffman

Laurie Huffman

Numerade Educator

Problem 2

Find the dimensions of the given matrix and identify the given entry.
$$
B=\left[\begin{array}{ll}
44 & 55
\end{array}\right] ; b_{12}
$$

Laurie Huffman

Numerade Educator

Problem 3

Find the dimensions of the given matrix and identify the given entry.
$$
C=\left[\begin{array}{r}
\frac{5}{2} \\
1 \\
-2 \\
8
\end{array}\right] ; C_{11}
$$

Laurie Huffman

Numerade Educator

Problem 4

Find the dimensions of the given matrix and identify the given entry.
$$
D=\left[\begin{array}{rr}
15 & -18 \\
6 & 0 \\
-6 & 5 \\
48 & 18
\end{array}\right] ; d_{31}
$$

Laurie Huffman

Numerade Educator

Problem 5

Find the dimensions of the given matrix and identify the given entry.
$$
E=\left[\begin{array}{ccccc}
e_{11} & e_{12} & e_{13} & \ldots & e_{1 q} \\
e_{21} & e_{22} & e_{23} & \ldots & e_{2 q} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
e_{p 1} & e_{p 2} & e_{p 3} & \ldots & e_{p q}
\end{array}\right] ; E_{22}
$$

Laurie Huffman

Numerade Educator

Problem 6

Find the dimensions of the given matrix and identify the given entry.
$$
A=\left[\begin{array}{rrr}
2 & -1 & 0 \\
3 & 5 & -3
\end{array}\right] ; A_{21}
$$

Laurie Huffman

Numerade Educator

Problem 7

Find the dimensions of the given matrix and identify the given entry.
$$
B=\left[\begin{array}{rr}
1 & 3 \\
5 & -6
\end{array}\right] ; b_{12}
$$

Laurie Huffman

Numerade Educator

Problem 8

Find the dimensions of the given matrix and identify the given entry.
$$
C=\left[\begin{array}{cccc}
x & y & w & e \\
z & t+1 & 3 & 0
\end{array}\right] ; C_{23}
$$

Laurie Huffman

Numerade Educator

Problem 9

Find the dimensions of the given matrix and identify the given entry.
$$
D=\left[\begin{array}{llll}
d_{1} & d_{2} & \ldots & d_{n}
\end{array}\right] ; D_{1 r}(\text { any } r)
$$

Laurie Huffman

Numerade Educator

Problem 10

Find the dimensions of the given matrix and identify the given entry.
$$
\left.E=\left[\begin{array}{llll}
d & d & d & d
\end{array}\right] ; E_{1 r} \text { (any } r\right)
$$

Laurie Huffman

Numerade Educator

Problem 11

Solve for $x, y, z$, and $w$.
$$
\left[\begin{array}{cc}
x+y & x+z \\
y+z & w
\end{array}\right]=\left[\begin{array}{ll}
3 & 4 \\
5 & 4
\end{array}\right]
$$

Laurie Huffman

Numerade Educator

Problem 12

Solve for $x, y, z$, and $w$.
$$
\left[\begin{array}{cc}
x-y & x-z \\
y-w & w
\end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\
0 & 6
\end{array}\right]
$$

Laurie Huffman

Numerade Educator

Problem 13

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
A+B
$$

Laurie Huffman

Numerade Educator

Problem 14

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
A-C
$$

Laurie Huffman

Numerade Educator

Problem 15

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
A+B-C
$$

Laurie Huffman

Numerade Educator

Problem 16

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
12 B
$$

Laurie Huffman

Numerade Educator

Problem 17

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
2 A-C
$$

Laurie Huffman

Numerade Educator

Problem 18

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$

Laurie Huffman

Numerade Educator

Problem 19

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
2 A^{T}
$$

Laurie Huffman

Numerade Educator

Problem 20

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\1 & 0 \\-1 & 2
\end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\0 & 0.5 \\-1 & 3\end{array}\right], \text { and } \\&C=\left[\begin{array}{rr}1 & -1 \\1 & 1 \\-1 & -1\end{array}\right].\end{aligned}$$
$$
A^{T}+3 C^{T}
$$

Laurie Huffman

Numerade Educator

Problem 21

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
A+B
$$

Laurie Huffman

Numerade Educator

Problem 22

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
B-C
$$

Laurie Huffman

Numerade Educator

Problem 23

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
A-B+C
$$

Laurie Huffman

Numerade Educator

Problem 24

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
\frac{1}{2} B
$$

Laurie Huffman

Numerade Educator

Problem 25

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
2 A-B
$$

Laurie Huffman

Numerade Educator

Problem 26

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
2 A-4 C
$$

Laurie Huffman

Numerade Educator

Problem 27

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
3 B^{T}
$$

Laurie Huffman

Numerade Educator

Problem 28

Evaluate the given expression. Take $A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]$, and $C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .$
$$
2 A^{T}-C^{T}
$$

Laurie Huffman

Numerade Educator

Problem 29

Use technology.
$$
A-C
$$

Laurie Huffman

Numerade Educator

Problem 30

Use technology.
$$
C-A
$$

Laurie Huffman

Numerade Educator

Problem 31

Use technology.
$$
1.1 B
$$

Laurie Huffman

Numerade Educator

Problem 32

Use technology.
$$
-0.2 B
$$

Laurie Huffman

Numerade Educator

Problem 33

Use technology.
$$
A^{T}+4.2 B
$$

Laurie Huffman

Numerade Educator

Problem 34

Use technology.
$$
(A+2.3 C)^{T}
$$

Laurie Huffman

Numerade Educator

Problem 35

Use technology.
$$
(2.1 A-2.3 C)^{T}
$$

Laurie Huffman

Numerade Educator

Problem 36

Use technology.
$$
(A-C)^{T}-B
$$

Laurie Huffman

Numerade Educator

Problem 37

The following table shows the number of Mac computers, iPods, and iPhones sold, in millions of units, in the fourth quarter (Q4) of 2006 , and the changes over the previous year in each of $2007 \mathrm{Q} 4$ and $2008 \mathrm{Q} 4$.
$$
\begin{array}{|r|c|c|c|}
\hline & \text { Macs } & \text { iPods } & \text { iPhones* } \\
\hline \text { Q4 2006 } & 1.6 & 8.8 & 0 \\
\hline \text { Change in 2007 } & 0.6 & 1.5 & 1.1 \\
\hline \text { Change in 2008 } & 0.4 & 0.9 & 5.8 \\
\hline
\end{array}
$$

Lauren Shelton

Numerade Educator

Problem 38

The following table shows annualized spending, in billions of dollars, on residential, office, and educational construction in the United States in August 2008 , as well as the month-over-month changes for September and October. ${ }^{2}$.
$$
\begin{array}{|c|c|c|c|}
\hline & \text { Residential } & \text { Office } & \text { Educational } \\
\hline \text { August } & 361 & 74 & 106 \\
\hline \text { Change in September } & -1 & 1 & 0 \\
\hline \text { Change in October } & -13 & 2 & 1 \\
\hline
\end{array}
$$

Lauren Shelton

Numerade Educator

Problem 39

The Left Coast Bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of book: hardcover, softcover, and plastic (for infants). At the beginning of January, the central computer showed the following books in stock:
$$
\begin{array}{|r|c|c|c|}
\hline & \text { Hard } & \text { Soft } & \text { Plastic } \\
\hline \text { San Francisco } & 1,000 & 2,000 & 5,000 \\
\hline \text { Los Angeles } & 1,000 & 5,000 & 2,000 \\
\hline
\end{array}
$$
Suppose its sales in January were as follows: 700 hardcover books, 1,300 softcover books, and 2,000 plastic books sold in San Francisco, and 400 hardcover, 300 softcover, and 500 plastic books sold in Los Angeles. Write these sales figures in the form of a matrix, and then show how matrix algebra can be used to compute the inventory remaining in each store at the end of January.

Lauren Shelton

Numerade Educator

Problem 40

The Left Coast Bookstore chain discussed in Exercise 39 actually maintained the same sales figures for the first 6 months of the year. Each month, the chain restocked the stores from its warehouse by shipping 600 hardcover, 1,500 softcover, and 1,500 plastic books to San Francisco and 500 hardcover, 500 softcover, and 500 plastic books to Los Angeles.
a. Use matrix operations to determine the total sales over the 6 months, broken down by store and type of book.
b. Use matrix operations to determine the inventory in each store at the end of June.

Lauren Shelton

Numerade Educator

Problem 41

Annual revenues and production costs at Luddington's Wellington Boots \& Co. are shown in the following spreadsheet.
Use matrix algebra to compute the profits from each sector each year.

Barsha Rana

Numerade Educator

Problem 42

The following spreadsheet gives annual production costs and profits at Gauss-Jordan Sneakers, Inc.
Use matrix algebra to compute the revenues from each sector each year.

Barsha Rana

Numerade Educator

Problem 43

In 1980 the U.S. population, broken down by regions, was $49.1$ million in the Northeast, $58.9$ million in the Midwest, $75.4$ million in the South, and $43.2$ million in the West. ${ }^{3}$ In 1990 the population was $50.8$ million in the Northeast, $59.7$ million in the Midwest, $85.4$ million in the South, and $52.8$ million in the West. Set up the population figures for each year as a row vector, and then show how to use matrix operations to find the net increase or decrease of population in each region from 1980 to 1990 .

Barsha Rana

Numerade Educator

Problem 44

In 1990 the U.S. population, broken down by regions, was $50.8$ million in the Northeast, $59.7$ million in the Midwest, $85.4$ million in the South, and $52.8$ million in the West. ${ }^{4}$ Between 1990 and 2000, the population in the Northeast grew by $2.8$ million, the population in the Midwest grew by $4.7$ million, the population in the South grew by $14.8$ million, and the population in the West grew by $10.4$ million. Set up the population figures for 1990 and the growth figures for the decade as row vectors. Assuming that the population will grow by the same numbers from 2000 to 2010 as they did from 1990 to 2000 , show how to use matrix operations to find the population in each region in 2010 .

Barsha Rana

Numerade Educator

Problem 45

Use matrix algebra to determine the total number of foreclosures in each of the given months.
${ }^{3}$ Source: U.S. Census Bureau, Statistical Abstract of the United States:
2001 (www.census.gov).
${ }^{4}$ Ibid.
${ }^{5}$ Data are rounded to the nearest 100 . Source for data:
www.currentforeclosures.com/Stats/

Lauren Shelton

Numerade Educator

Problem 46

Use matrix algebra to determine the total number of foreclosures in each of the given states during the last four months of 2008 .

Lauren Shelton

Numerade Educator

Problem 47

Use matrix algebra to determine in which month the difference between the number of foreclosures in California and in Florida
was greatest.

Lauren Shelton

Numerade Educator

Problem 48

Use matrix algebra to determine in which region the difference between the foreclosures in August and December was greatest.

Barsha Rana

Numerade Educator

Problem 49

Microbucks Computer Company makes two computers, the Pomegranate II and the Pomegranate Classic, at two different factories. The Pom II requires 2 processor chips, 16 memory chips, and 20 vacuum tubes, while the Pom Classic requires 1 processor chip, 4 memory chips, and 40 vacuum tubes. Microbucks has in stock at the beginning of the year 500 processor chips, 5,000 memory chips, and 10,000 vacuum tubes at the Pom II factory, and 200 processor chips, 2,000 memory chips, and 20,000 vacuum tubes at the Pom Classic factory. It manufactures 50 Pom II's and 50 Pom Classics each month.
a. Find the company's inventory of parts after two months, using matrix operations.
b. When (if ever) will the company run out of one of the parts?

Barsha Rana

Numerade Educator

Problem 50

Microbucks Computer Company, besides having the stock mentioned in Exercise 49 , gets shipments of parts every month in the amounts of 100 processor chips, 1,000 memory chips, and 3,000 vacuum tubes at the Pom II factory, and 50 processor chips, 1,000 memory chips, and 2,000 vacuum tubes at the Pom Classic factory.
a. What will the company's inventory of parts be after six months?
b. When (if ever) will the company run out of one of the parts?

Barsha Rana

Numerade Educator

Problem 51

The following table gives the number of people (in thousands) who visited Australia and South Africa in $1998^{6}$ :
$$
\begin{array}{|l|c|c|}
\hline & \text { To } & \text { Australia } & \text { South Africa } \\
\hline \text { From } & \text { North America } & 440 & 190 \\
\hline \text { Europe } & 950 & 950 \\
\hline \text { Asia } & 1,790 & 200 \\
\hline
\end{array}
$$
You predict that in $2008,20,000$ fewer people from North America will visit Australia and 40,000 more will visit South Africa, 50,000 more people from Europe will visit each of Australia and South Africa, and 100,000 more people from Asia will visit South Africa, but there will be no change in the number visiting Australia.
a. Use matrix algebra to predict the number of visitors from the three regions to Australia and South Africa in 2008 .
b. Take $A$ to be the $3 \times 2$ matrix whose entries are the 1998
tourism figures and take $B$ to be the $3 \times 2$ matrix whose entries are the 2008 tourism figures. Give a formula (in terms of $A$ and $B$ ) that predicts the average of the numbers of visitors from the three regions to Australia and South Africa in 1998 and $2008 .$ Compute its value.

Carson Merrill

Numerade Educator

Problem 52

Referring to the 1998 tourism figures given in the preceding exercise, assume that the following (fictitious) figures represent the corresponding numbers from 1988 .
$$
\begin{array}{|l|c|c|}
\hline & \text { To } & \text { Australia } & \text { South Africa } \\
\hline \text { From } & \text { North America } & 500 & 100 \\
\hline \text { Europe } & 900 & 800 \\
\hline \text { Asia } & 1,400 & 50 \\
\hline
\end{array}
$$
Take $A$ to be the $3 \times 2$ matrix whose entries are the 1998 tourism figures and take $B$ to be the $3 \times 2$ matrix whose entries are the 1988 tourism figures.
a. Compute the matrix $A-B$. What does this matrix represent?
b. Assuming that the changes in tourism over $1988-1998$ are repeated in $1998-2008$, give a formula (in terms of $A$ and $B$ ) that predicts the number of visitors from the three regions to Australia and South Africa in 2008 .

Carson Merrill

Numerade Educator

Problem 53

Is it possible for $a 2 \times 3$ matrix to equal a $3 \times 2$ matrix? Explain.

Mahendra Kumar

Numerade Educator

Problem 54

Is it possible for $a 2 \times 3$ matrix to equal a $3 \times 2$ matrix? Explain.

Mahendra Kumar

Numerade Educator

Problem 55

If $A$ and $B$ are $2 \times 3$ matrices and $A=B$, what can you say about $A-B ?$ Explain.

Mahendra Kumar

Numerade Educator

Problem 56

What does it mean when we say that $(A+B)_{i j}=A_{i j}+B_{i j}$ ?

Mahendra Kumar

Numerade Educator

Problem 57

What would a $5 \times 5$ matrix $A$ look like if $A_{i i}=0$ for every $i$ ?

Mahendra Kumar

Numerade Educator

Problem 58

What would a matrix $A$ look like if $A_{i j}=0$ whenever $i \neq j$ ?

Mahendra Kumar

Numerade Educator

Problem 59

Give a formula for the $i j$ th entry of the transpose of a $\operatorname{matrix} A$.

Laurie Huffman

Numerade Educator

Problem 60

A matrix is symmetric if it is equal to its transpose. Give an example of a. a nonzero $2 \times 2$ symmetric matrix and b. a nonzero $3 \times 3$ symmetric matrix.

Mahendra Kumar

Numerade Educator

Problem 61

A matrix is skew-symmetric or antisymmetric if it is equal to the negative of its transpose. Give an example of a. a nonzero $2 \times 2$ skew-symmetric matrix and $\mathbf{b}$. a nonzero $3 \times 3$ skew-symmetric matrix.

Mahendra Kumar

Numerade Educator

Problem 62

Referring to Exercises 60 and 61 , what can be said about a matrix that is both symmetric and skew-symmetric?

Mahendra Kumar

Numerade Educator

Problem 63

Why is matrix addition associative?

Mahendra Kumar

Numerade Educator

Problem 64

Describe a scenario (possibly based on one of the preceding examples or exercises) in which you might wish to compute $A-2 B$ for certain matrices $A$ and $B$.

Mahendra Kumar

Numerade Educator

Problem 65

Describe a scenario (possibly based on one of the preceding examples or exercises) in which you might wish to compute $A+B-C$ for certain matrices $A, B$, and $C .$

Mahendra Kumar

Numerade Educator