It has been said that sports fans are the nation’sforemost consumers of statistics and that baseball fansare the most prominent among them. Whether it’s in
the information conveyed by a scoreboard or in the currentinformation on league leaders, baseball records are full ofnumbers.
• How can large collections of data be organized andmanaged in an efficient way?
• What calculations provide meaningful information topeople who use the data?
• How can computers and calculators assist them?
Baseball statisticians, business executives, and wildlifebiologists are among the diverse groups of people who turn to the mathematics of matrices for answers to thesequestions.
Matrix Operations 3and Applications
CHAPTER
Lesson 3.1Addition and Subtraction of Matrices
Lesson 3.2Multiplication of Matrices, Part 1
Lesson 3.3Multiplication of Matrices, Part 2
Lesson 3.4Population Growth: The Leslie Model, Part 1
Lesson 3.5Population Growth: The Leslie Model, Part 2
In Lesson 1.2, you were introduced to matrices as a natural way to organize,manipulate, and display information. As you have seen, matrices can beused to represent sets of discrete data that can be described with twocharacteristics. One characteristic can be represented by the rows of thematrix and the other by the columns. In this lesson you will be introducedto some of the terminology and notation used when working with matrices.Matrix addition and subtraction will also be explored.
Matrix TerminologySuppose that you and a few of your friends are planning a pizza and videoparty. You decide to call several pizza houses to ask about prices for largesingle-topping pizzas, liter containers of cold drinks, and family-sizedsalads with house dressing. You could record your information in a tablesuch as the following.
121Lesson 3.1 • Addition and Subtraction of Matrices
Or you might choose to organize your data in matrix form by writingthe numbers in a rectangular array and enclosing them in brackets orparentheses.
When writing this price matrix, you could omit the rowand column labels and dollar signs and write only the values.However, if you delete the labels you will have to rememberthat the rows represent the prices for pizzas, drinks, andsalads, while the columns represent the various pizza houses.
This matrix has 3 rows and 4 columns. So its order ordimension is 3 by 4 (written as 3 × 4). Notice that when yougive the order of a matrix, you write the number of rowsfollowed by the number of columns. Each of the individualentries in the matrix is called an element or a component ofthe matrix.
In general, a matrix with m rows and n columns iscalled an m by n matrix.
After looking over your data, you might decide to dropGina’s options since they are more expensive than any of theother pizza houses. If you do this, you will be left with a 3 × 3square matrix. Notice that in a square matrix the number ofrows equals the number of columns or m = n.
122 Chapter 3 • Matrix Operations and Applications
If a matrix has only one column, it is called a column matrix. Forexample, if you list only the prices for Sal’s offerings, the result will be acolumn matrix of order 3 × 1.
If you choose to look at the pizza pricesalone, they can be represented with a 1 × 3row matrix.
Vin’s Toni’s Sal’sPizza [$10.10 $10.86 $10.65]
Exercises1. Give an example of a 2 × 5 matrix, a row matrix with two
elements, and a square matrix.
2. How many elements does a 2 × 8 matrix have? A 7 × 1 matrix? Anm × n matrix?
3. A trendy garment company receives orders from three clothingshops. The first shop orders 25 jackets, 75 shirts, and 75 pairs ofpants. The second shop orders 30 jackets, 50 shirts, and 50 pairs ofpants. The third shop orders 20 jackets, 40 shirts, and 35 pairs ofpants. Display this information in a matrix. Let the rows representthe shops and the columns represent the type of garment ordered.Label the rows and columns of your matrix accordingly.
4. Although matrices contain many data values, they can also bethought of as single entities. This feature allows us to refer to amatrix with a single capital letter.
123Lesson 3.1 • Addition and Subtraction of Matrices
Individual entries in a matrix are identified by row number andcolumn number, in that order. For example, the value $10.65 is theentry in row 1 and column 3 of matrix A and is referenced as A13.Entry A13 represents or is interpreted as the cost of a pizza at Sal’s.Notice that A31 is not the same as A13. Entry A31 has the value $3.69and represents the cost of a salad at Vin’s.
a. What is the value of A21? Of A12? Of A32?
b. Write an interpretation for entry A21. For entry A12. For entry A32.
5. In the row matrix S, the entries are referenced as S1, S2, and S3.
Write an interpretation of S3.
6. For breakfast Yoko had cereal, a banana, a cup of milk, and a sliceof toast. She recorded the following information in her foodjournal. Cereal: 165 calories, 3 g fat, 33 g carbohydrate, and nocholesterol. Banana: 120 calories, no fat, 26 g carbohydrate, andno cholesterol. Milk: 120 calories, 5 g fat, 11 g carbohydrate, and15 mg cholesterol. Toast: 125 calories, 6 g fat, 14 g carbohydrate,and 18 mg cholesterol.
a. Write this information in a matrix N whose rows represent thefoods. Label the rows and columns of your matrix.
b. State the values of N23, N32, and N42.
c. Write an interpretation of N23, N32, and N42.
7. As you continue to plan your pizza party, you discover that the localsupermarket has a sale on 2-liter bottles of soft drinks. You decidenot to order drinks from a pizza house after all. Write and label a 2 × 3 matrix that represents the prices for just pizza and salad atVin’s, Toni’s, and Sal’s.
124 Chapter 3 • Matrix Operations and Applications
8. Suppose that when you were calling the pizza houses about prices,you also collected the following information about the cost ofadditional toppings and salad dressings.
Represent the information from this table in another 2 × 3 matrixwhose rows represent the additional toppings and dressings andwhose columns represent the three pizza houses. Label the rowsand columns of your matrix.
9. Suppose you want to find the cost of ordering pizzas with twotoppings and salads with a choice of two salad dressings. This canbe done by adding corresponding elements of your two pricematrices from Exercises 7 and 8. If you let A represent the basicprice matrix and B represent the matrix of additional costs, thenyou can add A and B to get a third matrix C. Matrix C will representthe total prices for pizza and salads at each pizza house.
125Lesson 3.1 • Addition and Subtraction of Matrices
10. In Exercise 9, the entries of matrix C represent the sum of thecorresponding entries in matrices A and B. For example, C13, whichrepresents the cost of a pizza with an extra topping at Sal’s, equalsthe sum of A13 and B13.
a. What is the value of A21? Of B21? Of C21?
b. Write an interpretation of A21, B21, and C21.
In Exercises 9 and 10, you found that you could add two matrices, A andB, by adding the corresponding entries of each. In general, you can addor subtract matrices only if they have the same orders.
If A and B are matrices of the same orders, thenthe sum A + B is formed by adding thecorresponding entries of A and B. The differenceA – B is formed by subtracting the correspondingentries of B from the corresponding entries of A.
11. Find the value of each of the following expressions. If it is notpossible, explain.
Find and label a matrix that displays the changes in these statisticsfrom the 2011 season to the 2012 season. Notice that several of thestatistics decreased from 2011 to 2012. How will you show this inyour matrix?
13. The matrices that follow give the winning times in seconds for threetrack and field events in the 1964 and 2012 Olympic Games.
100-meter race200-meter race400-meter race
a. Find and label the matrix that represents the change in times inseconds for each event from 1964 to 2012.
b. In which event and sex was there the greatest decrease in time?The smallest decrease?
14. In your study of algebra, you learned that the commutative and associative properties hold for addition over the set of realnumbers, that is, for all real numbers a, b, and c, a + b = b + aand a + (b + c) = (a + b) + c.
127Lesson 3.1 • Addition and Subtraction of Matrices
a. Do you think that the commutative and associative propertieshold for addition of matrices? Why?
b. Use the following matrices to test your conjecture in part a.
A = B = C =
15. Do you think that the commutative and associative properties holdfor subtraction of matrices? Why? Test your conjecture usingmatrices A, B, and C in Exercise 14b.
16. A matrix all of whose entries are the number zero is called a zeromatrix and is denoted using a capital letter O alone or withsubscripts Om×n.
a. Use matrix A from Exercise 14b to show that A + O = O + A = Aand that A – A = O.
b. Show that A + (–A) = (–A) + A = O, where the matrix –A, calledthe negative of A, is obtained by negating each entry in A.
17. The matrix M below shows the mileage between 5 major U.S. cities.
a. Entries that are located in row i, column j,where i = j, are said to be located on the maindiagonal of the matrix. Examine the entrieson the main diagonal of M. What do younotice?
b. A square matrix R with order n × n issymmetric if Rij = Rji, where i and j = 1, 2, 3, . . ., n. Is matrix M symmetric? Explain.
c. Give an example of a 3 × 3 matrix that issymmetric.
d. Could a matrix that is not square besymmetric? Why?
2 41 1
� �−
1 32 5−
4 23 1
–� �
In a symmetric matrix youneed to know only the valuesalong the main diagonal andeither the triangle above themain diagonal (the uppertriangle) or below it (the lowertriangle). Because of thisfeature, symmetric matricesare often written with blanksin either the upper or lowertriangle.
128 Chapter 3 • Matrix Operations and Applications
18. In statistics, a correlation matrix is a matrix whose entriesrepresent the degree of relationship between variables. The valuesin a correlation matrix range from –1 to 1, where 0 indicates thatthere is no relationship, a negative value indicates that as onevariable increases the other decreases, and a positive valueindicates that as one variable increases the other one alsoincreases. In a study of the relationship between ACT test scores,high school class rank, and college grade point average, thefollowing correlation matrix was generated. Notice that the rowlabels and the column labels are the same in a correlation matrix.
Source: Aksamit, Mitchell, and Pozehl, 1986.
a. Why do you think the values along the main diagonal of acorrelation matrix are all 1s?
b. Why are the values in a correlation matrix symmetric about themain diagonal?
c. Which variable had the highest correlation with college GPA?
d. Which subject area test had the highest correlation with highschool rank?
Project19. Bring to class at least two matrices from newspapers or magazines.
Be prepared to share your matrices with other class members.
a. What are the dimensions of each of your matrices?
b. What is represented by the rows and columns of your matrices?
In the previous lesson, matrix addition and subtraction were defined bylooking at several matrix models of real-world situations. In this lesson,matrix multiplication is approached in a similar manner.
Multiplying a Matrix by a ScalarReturn to the pizza problem from the last lesson. In that lesson you usedthe following price matrix to represent the costs of two-topping pizzas andsalads with a choice of two dressings from Vin’s, Toni’s, and Sal’s.
C =
Suppose you wish to order four of each of the pizzas and four of eachof the salads. To do this, multiply each element in matrix C by 4 to get anew matrix that is equal to 4C.
130 Chapter 3 • Matrix Operations and Applications
If you call the new matrix T and label the rows and columns of thematrix, you have
T = .
When working with matrices, a real number is often called a scalar.
In general, if k is a real number and A is a matrix,the matrix kA is formed by multiplying each entryin the matrix A by k.
Multiplying a Column Matrix by a Row MatrixConsider the following situation.
Suppose that Ruben, a student at Washington High, goes to a nearbystore to buy some food to stock up his locker for between-class snacks. Hechooses four small bags of chips, five candy bars, a box of cheese crackers,three packs of sour drops, and two bags of cookies.
Ruben’s purchases can be represented by a row matrix Q.
Suppose further that chips cost 30 cents a bag, candy bars cost 35cents each, crackers cost 50 cents a box, sour drops cost 20 cents a pack,and cookies sell for 75 cents a bag. These prices can be represented incolumn matrix P.
P =
Now the obvious question to ask is, “How much did Ruben pay for allthese snacks?” You can answer this question by multiplying the pricematrix P by the quantity matrix Q.
131Lesson 3.2 • Multiplication of Matrices, Part 1
Q × P = [4 5 1 3 2]
= 4(30) + 5(35) + 1(50) + 3(20) + 2(75)
= 120 + 175 + 50 + 60 + 150
= 555 cents = $5.55
This matrix computation is, of course, exactly what the clerk at thestore would do in figuring Ruben’s bill. The price of each item is multipliedby the number purchased and the products are summed.
In order to do this computation, the number of entries in each matrixmust be the same. Items and prices must also correspond.
In general, if Q is a row matrix and P is a columnmatrix, each having the same number of entries, thenthe product QP is defined. QP can be determined bymultiplying the corresponding entries and summingthe results.
ExampleSuppose a second student, Terri, goes along with Ruben to the store. Herpurchases are a bag of chips, two candy bars, two packs of gum that cost25 cents each, and a medium drink for 75 cents. Find the total cost ofTerri’s purchases.
Solution:
To solve this problem, use a row matrix to represent the quantity ofeach item and a column matrix to represent the price of each item.
132 Chapter 3 • Matrix Operations and Applications
Let P represent the item-price matrix.
P =
To find the total cost of the purchase find the product QP.
Q × P = [1 2 2 1]
= 1(30) + 2(35) + 2(25) + 1(75)
= 30 + 70 + 50 + 75
= 225 cents = $2.25
As you can see from these examples, if a (k × 1) column matrix P ismultiplied by a (1 × k) row matrix Q, the result is a (1 × 1) single-valuematrix.
Multiplying a Matrix with More than OneColumn by a Row MatrixNow that you can multiply a column matrix by a row matrix, you can usethat process to multiply a multidimensional matrix by a row matrix.
Return to the pizza problem. Suppose your group decides to orderfive pizzas and three salads and you want to calculate the total cost ateach of the pizza houses. If you do the calculations without usingmatrices, you multiply the pizza price by 5 and add the result to 3 timesthe salad price for each pizza house.
133Lesson 3.2 • Multiplication of Matrices, Part 1
Set up a 2 × 3 matrix C to represent the prices for the pizzas andsalads at each of the three pizza houses.
C =
Now when matrix C is multiplied by the row matrix A, the expectedresult is another matrix whose entries will give the total cost for five pizzasand three salads at each pizza house. To accomplish this, it makes senseto multiply each of the columns in matrix C by the row matrix A.
The last step is to label the entries in the final product:
Notice that in carrying out this matrix computation, the process wasexactly the same as that of calculating the costs without the use ofmatrices. In the matrix multiplication, each of the entries in the columnsof matrix C was multiplied by the corresponding entry in the row matrixA. These products were then summed to give the entries of the finalproduct matrix.
You can see from this model that multiplying a multidimensionalmatrix by a row matrix can only be defined if the number of entries in therow matrix equals the number of rows in the multidimensional matrix.
In general, the product of a (1 × k) row matrix Aand a (k × n) matrix C is a (1 × n) row matrix P.
Exercises1. Recall that a matrix T was found by multiplying 4 times the price
matrix that represented the costs of two-topping pizzas and saladswith a choice of two dressings from Vin’s, Toni’s, and Sal’s. (Seepage 130.)
T =
a. What does matrix T represent?
b. What is the cost of four pizzas at Sal’s?
c. Interpret T12 and T21.
2. A jeweler has a small shop where she makes and sells four differentkinds of jewelry: earrings (e), pins (p), necklaces (n), and bracelets(b). She makes each item out of either pearls or jade beads. Thefollowing matrix represents the jeweler’s sales for May.
M =
The jeweler hopes to sell twice as many of each piece in June.
135Lesson 3.2 • Multiplication of Matrices, Part 1
a. Calculate a matrix J, where J = 2M to represent the number ofeach item the jeweler will sell in June if she reaches her goal.
b. Label the rows and columns of matrix J.
c. How many jade necklaces does the jeweler expect to sell in June?
d. Interpret J21 and J12.
3. Matt reads on the side of his cereal box that each ounce of cerealcontains the following percentages of the minimum dailyrequirements of:
Vitamin A 25%Vitamin C 25%Vitamin D 10%
If Matt eats 3 ounces of cereal for breakfast, what percentages ofeach vitamin will he get? Show the matrices and matrix operationinvolved in your calculation. Label your matrices.
4. The regents at a state university recently announced a 7% raise oftuition rates per semester hour. The current rates per semester hourare shown in the following table.
Undergraduate Graduate
Resident $75.00 $99.25Nonresident $204.00 $245.25
a. Write and label a matrix that represents this information.
b. Find a new matrix that represents the tuition rates per semesterhour after the 7% raise goes into effect. Label your matrix.
c. Find a matrix that represents the dollar increase for each of thecategories. Label your matrix.
5. For each of the following, state whether the matrix product QP isdefined. If so, give the order of the product.
136 Chapter 3 • Matrix Operations and Applications
6. A local credit union has investments in three states—Massachusetts,Nebraska, and California. The deposits in each state are dividedbetween consumer loans and bonds.
The amount of money (in thousands of dollars) invested in eachcategory is displayed in the following table.
MA NE CA
Loans 230 440 680Bonds 780 860 940
The current yields on these investments are 6.5% for consumerloans and 7.2% for bonds. Use matrix multiplication to find thetotal earnings for each state. Label your matrices.
7. A carpenter makes a trip to the lumber company to pick up ten 2 × 6s, four 4 × 6s, and two 5 × 5s. In 8-foot lengths, 2 × 6s cost$3.00, 4 × 6s cost $8.50, and 5 × 5s cost $9.50.
a. Write and label a row matrix and a column matrix to representthe information in this problem.
b. Will everyone necessarily write the same row and columnmatrices? Explain your answer.
c. Perform a matrix multiplication to find the total cost of thecarpenter’s purchases.
8. Use the following matrices to compute the given expressions.
A = B = [2 4] C =
a. 3A
b. BA
c. BC
d. –2C
9. You have $10,000 in a 12-month CD at 7.3% (annual yield); $17,000in a credit union at 6.5%; and $12,000 in bonds at 7.5%. Usemultiplication of a column matrix by a row matrix to find yourearnings for a year. Label your matrices.
137Lesson 3.2 • Multiplication of Matrices, Part 1
10. The transpose (AT) of a matrix A is the matrix obtained byinterchanging the rows and columns of matrix A.
a. Describe the transpose of a row matrix.
b. Describe the transpose of a column matrix.
c. Matrix M from Exercise 2 is given below.
M =
Write and label the transpose (MT) of matrix M.
11. Refer to Exercise 2. Suppose it takes the jeweler 2 hours to make apair of earrings, 1 hour to make a pin, 2.5 hours to make anecklace, and 1.5 hours to make a bracelet.
a. Write and label a row matrix that represents this information.
b. Use matrix multiplication to find a matrix that represents thetotal hours the jeweler spends making each type of jewelry(pearls or jade) for the month of May. (Hint: Use the transpose ofmatrix M that you found in the previous exercise.)
c. Label your product matrix.
d. Interpret each of the entries in the product matrix.
12. A hobby shop has three different locations, North, South, and East.The store’s sales for July are shown in the following table.
Suppose that model trains sell for $40 each, cars for $35, planes for $80, and trucks for $45. Use matrix multiplication to find theshop’s total sales at each location. Label your matrices.
138 Chapter 3 • Matrix Operations and Applications
13. During the first week of a recent fundraiser, math club studentssold the following number of calendars.
Mon Tues Wed Thurs Fri
Calendars 10 15 20 30 50
a. Write this information in a column matrix C. Label your matrix.
b. Find a row matrix N such that the product N times C gives thetotal number of calendars that the students sold for the week.Then use matrix multiplication to find the total number ofcalendars.
c. Find a row matrix A such that the product A times C gives theaverage number of calendars that the students sold each day.(Hint: What fraction would you multiply the total number ofcalendars by to find the average?) Then use matrixmultiplication to find the daily average
In Lesson 3.2 you multiplied a row matrix times a 2 by 3 price matrix todetermine the total cost of 5 pizzas and 3 salads at each pizza house. (Seepage 133.) But what if you want to compare the total costs of severaldifferent combinations of pizzas and salads?
One way to do this is to multiply each of the row matrices times aprice matrix and then compare the products. Another way is to combinethe combination options into a single matrix. Then you can multiply thisnew matrix times the price matrix. For example, let matrix B representthree different pizza/salad combinations and matrix C represent the pricematrix for the pizza houses.
B = C =
If you multiply matrix B times matrix C, the product will be a 3 × 3matrix (call it D). The rows of D will represent the three options and thecolumns will represent the three pizza houses. The elements of D give thetotal cost for each of the three options at each of the three pizza houses.
Notice as you follow the steps of this matrix multiplication that thecomputations are exactly the same as making three separate calculations,one for each option. You expect, then, that row 1 of the product representsthe cost of four pizzas and three salads, that row 2 of the productrepresents the cost of four pizzas and four salads, and that row 3 of theproduct represents the cost of five pizzas and three salads at each of thepizza houses.
141Lesson 3.3 • Multiplication of Matrices, Part 2
Notice that the number of rows of the first matrix and the number ofcolumns of the second give the order of the product.
It is important to observe that the dimensions of these matrices canalso be described using the row and column labels. Matrix B classifies thedata according to Options (rows) and Foods (columns). Hence you canrefer to matrix B as an Options by Foods matrix. Likewise you can describeC as a Foods by Houses matrix. The product B times C, in turn, results in amatrix of dimension Options by Houses. (See the following diagram.)
Using row and column labels in this manner helps determinewhether a matrix multiplication will result in a meaningful interpretationor, indeed, whether it will give you the results that you want.
ExampleM is a 2 × 3 matrix and N is a 4 × 2 matrix. Which of the products isdefined, MN or NM? Explain.
Solution:
The product MN is not defined as thenumber of columns in the first matrix is notequal to the number of rows in the secondmatrix.
The product NM is defined as thenumber of columns in the first matrix is equalto the number of rows in the second matrix.The product matrix will be a 4 by 3 matrix.
×k n
B C D=
× × ×(m k) ( ) n)(m
Same
Dimensions of product
B C D=×
Same
Dimensions of product
Options by Foods Foods by Houses Options by Houses
142 Chapter 3 • Matrix Operations and Applications
Exercises1. Use the following matrices to compute the given expression. If the
expression is not defined, give the reason.
A = B = C =
a. AB
b. BA
c. BC
d. CB
2. Mike, Liz, and Kate are heirs to an estate that consists of acondominium, a customized BMW, and choice season tickets to theNebraska Cornhusker football games, and for the purposes of fairdivision, they have submitted the bids shown in matrix E.
E =
The awarding of the items in the estate is indicated by matrix A.
A =
a. Find the matrix product P = EA. Label the rows and columns of P.
b. Write an interpretation of the entries in matrix P. (Refer toExercise 7 in Lesson 2.2, pages 67 and 68.)
3. Rosa and Max go out to eat at Sammy’s Drive Inn. Rosa orders aSammy’s special, fries, and a shake. Max has a cheeseburger, abaked potato with sour cream, and a shake. The approximatenumbers of calories, grams of fat, and milligrams of cholesterol ineach of these foods are represented in the following table.
a. Write a matrix Q that describes Rosa’s and Max’s orders, withthe columns representing the foods. Label the rows and columnsof this matrix.
b. Write a matrix C that represents the information in thepreceding table with the rows representing the foods. Label therows and columns of this matrix.
c. What are the dimensions of matrix Q and of matrix C?
d. What is the dimension of the product Q times C? Show why youranswer is correct by using a diagram such as the one on page141.
e. The dimension of matrix Q could be described as Persons byFoods. Describe the dimensions of matrices C and Q times C in asimilar manner. Justify your answer for matrix Q times C with adiagram such as the one on page 141.
f. Multiply matrix Q times matrix C to get a matrix R. Label therows and columns of matrix R.
g. Interpret R12, R21, and R23.
4. a. What must be true about the dimensions of matrices A and B ifthe product C = AB is defined?
b. If the products AB and BA are both defined, what must be trueabout the dimensions of matrices A and B? Why?
c. Find two nonsquare matrices A and B, where AB and BA are bothdefined. Compute AB and BA. Does AB = BA? Why?
d. As illustrated by your answer in part c, if AB and BA are bothdefined, it does not necessarily follow that AB = BA (i.e., ingeneral, matrix multiplication is not commutative). Using 2 × 2matrices, find examples in which AB = BA and in which AB is not equal to BA.
144 Chapter 3 • Matrix Operations and Applications
5. An identity matrix is any matrix in which each entry along themain diagonal is 1 and all other entries are 0s. Identity matricesact in the same way for matrix products as the number 1 does fornumber products.
Let A be any 3 × 3 matrix and let
I = .
Show that IA = AI = A.
6. Given the matrices A, B, and C.
A = B = C =
a. Do you think that A(BC) = (AB)C?
b. Test your conjecture by computing the products A(BC) and (AB)C.
c. The computations in part b show one case in which matrixmultiplication is associative. Do you think this property holds forall matrices A, B, and C for which the product A(BC) is defined?Why or why not?
7. Find two (2 × 2) matrices A and B to demonstrate that (A + B)(A – B)is not necessarily equal to A2 – B2.
In algebra you learned that two numbers whose product is 1 (the identityelement for multiplication) are called inverses of each other. For
example, 5 and (or 5–1) are inverses of each other since 5 = 5 = 1.
Similarly, if A and B are two square matrices such that AB = BA = I, then Aand B are called inverses of each other. The inverse of A is denoted A–1.
8. a. Verify that the matrices A and B are inverses of each other bycomputing AB and BA.
145Lesson 3.3 • Multiplication of Matrices, Part 2
b. Not all square matrices will have an inverse. Use algebra to showthat matrix C does not have an inverse.
C =
9. Carefully plot the points A(0, 0), B(6, 2), C(8, 6), and D(2, 4) ongraph paper. Connect the points to form a polygon ABCD. You canrepresent this polygon with a matrix P as follows.
P =
a. Multiply the matrix that represents polygon ABCD by the matrix
T1 = .
b. Plot and label the four points represented in your new matrix asA’, B’, C’, and D’. Connect the points to form polygon A’B’C’D’.
c. Describe the relationship between polygon A’B’C’D’ and polygonABCD.
d. Multiply the matrix representing polygon A’B’C’D’ by the matrix
T2 = .
e. Plot and label the four points represented in your new matrix asA”, B”, C”, and D”. Describe the relationship between polygonA”B”C”D” and polygon A’B’C’D’.
f. Multiply T2T1 to get a new matrix R. Multiply R times the matrixP, that represents the original polygon ABCD, and plot theresulting points. What effect does multiplication by R have onABCD? Do the following to test your conjecture: Use a blank sheetof unlined paper and trace both your axes and polygon ABCD.Leave your copy on top of the original polygon and place thepoint of your pencil on the origin. Now, holding the originalpaper in place, rotate the top sheet until your copy of ABCDrests on top of polygon A”B”C”D”. Describe what happened topolygon ABCD.
146 Chapter 3 • Matrix Operations and Applications
g. Find a matrix T3 that reflects polygon A”B”C”D” about the y-axisinto quadrant IV of your graph.
h. Find a matrix T4 that rotates polygon A’B’C’D’ about the origininto quadrant IV. How does T4 relate to T2 and T3?
For Exercises 10–13, you need either a graphing calculator or access tocomputer software that performs matrix operations.
10. A manufacturing company that makes fine leather bags has threefactories—one in New York, one in Nebraska, and one in California.One of the bags they make comes in three styles—handbag,standard shoulder bag, and roomy shoulder bag. The production ofeach bag requires three kinds of work—cutting the leather, stitchingthe bag, and finishing the bag.
Matrix T gives the time (in hours) of each type of work required tomake each type of bag.
T =
Matrix P gives daily production capacity at each of the factories.
P =
Matrix W provides the hourly wages of the different workers ateach factory.
W =
Matrix D contains the total orders received at each factory for themonths of May and June.
147Lesson 3.3 • Multiplication of Matrices, Part 2
D =
a. Matrix T can be described as a Bag by Work matrix. Describematrices P, W, and D in a similar manner.
For parts b–e, use the matrices above (or their transposes). Label the rowsand columns of the matrix in each answer. Hint: The label dimensionsfrom part a will help you decide what your matrix products should look like.
b. Find the hours of each type of work needed each month to fill allorders.
c. Find the production cost per bag at each factory.
d. Find the cost of filling all May orders at the Nebraska factory.(Hint: In this example the answer, a single value, is the productof a row matrix and a column matrix).
e. Find the daily hours of each type of work needed at each factoryif production levels are at capacity.
11. (For students who have studied trigonometry.)
a. Plot the polygon ABCD represented in Exercise 9.
b. Multiply the matrix P by the following transformation matrix.
T1 =
c. Plot the resulting polygon and label it A’B’C’D’. How doespolygon A’B’C’D’ relate to polygon ABCD? Try repeating thetransformation using 180° to test your conjecture.
d. Write a matrix that will rotate a polygon through 60°. Does thistransformation matrix have the same effect as applying T1twice? Test your conjecture.
e. Find a matrix that rotates polygon ABCD through 90° andanother that rotates it through –90°. Find the product of thesetwo transformation matrices. What is the relationship betweenthese two matrices? Test your conjecture by finding the productof the matrices that will rotate the polygon through 60° and –60°.
148 Chapter 3 • Matrix Operations and Applications
12. The matrix A is called an upper-triangular matrix.
A =
a. Calculate A2, A3, and A4.
b. Make a conjecture about the form of Ak.
c. Test your conjecture by computing additional powers of A.
d. Challenge: Prove your conjecture using mathematical induction.
13. Challenge: Refer to Exercise 12 and explore the following.
a. Replace the 1s in the upper-triangular matrix A with 2s, 3s, and4s and repeat part a of Exercise 12 for each of your new upper-triangular matrices.
b. Use the results of part a to make a conjecture for Ak when the 1sin A are replaced by any natural number m.
c. Prove your conjecture in part b using mathematical induction.
Computer/Calculator Exploration14. Write a program for the graphing calculator based on the method
of Exercise 11 that will allow you to enter the coordinates of thevertices of a polygon and the angle of rotation. Design yourprogram so that both the original polygon and the rotation will be displayed.
Modeling Project15. Research and write a short report on modeling with matrices in
trigonometry. Possible topics include the representation vectors andcomplex numbers as matrices.
In previous lessons, you learned to add, subtract, and multiply matrices.In this lesson you will learn to use these skills to model the growth of apopulation.
Population growth is a topic that is of great concern to many people.For example, urban planners are interested in knowing how many peoplethere will be in various age groups after certain periods of time havepassed. Wildlife managers are concerned about keeping animalpopulations at levels that can be supported in their natural habitats.
If you know the age distribution of a population at a certain dateand the birth and survival rates for age-specific groups, you can use thisdata to create a mathematical model. You can use your model todetermine the age distributions of the survivors and descendants of theoriginal population at successive intervalsof time.
The problem used to illustrate thismodel was posed by P. H. Leslie. In hisproblem, the growth rate of a population ofan imaginary species of small brown rats,Rattus norvegicus, is examined.
Lesson 3.4
P. H. Leslie
In 1945, P. H. Leslie of the Bureau ofAnimal Population at Oxford Universityin Oxford, England created a powerfulmathematical model. This model, knownas the Leslie matrix model, is aneffective tool that can be used todetermine the growth of a population.
Suppose the original female rat population is 42 animals with theage distribution shown in the following table.
Initial Female Rat Population Age (months) 0–3 3–6 6–9 9–12 12–15 15–18
Number 15 9 13 5 0 0
You can use this table and the birth rate/survival rate information tofind the total number of rats and their age distribution after 3 months.
To find this new distribution, you will need to find the number of newfemale babies introduced into the population. You will also need todetermine the number of female rats that survive in each group and moveup to the next age group.
To find the number of new births after 3 months (1 cycle), multiplythe number of female rats in each age group times the correspondingbirth rates and then find the sum.
151Lesson 3.4 • Population Growth: The Leslie Model, Part 1
So after 3 months, there will be about 17 female rats in the 0–3 agegroup.
The number of female rats who survive in each age group and moveup to the next can be found as follows:
Age No. SR* Number Moving Up to the Next Age Group
0–3 15 0.6 (15)(0.6) = 9.0 move up to the 3–6 age group.3–6 9 0.9 (9)(0.9) = 8.1 move up to the 6–9 age group.6–9 13 0.9 (13)(0.9) =11.7 move up to the 9–12 age group.9–12 5 0.8 (5)(0.8) = 4.0 move up to the 12–15 age group.12–15 0 0.6 (0)(0.6) = 0 move up to the 15–18 age group.15–18 0 0 No rodent lives beyond 18 months.
* SR stands for survival rate.
The distribution of female rats after 3 months (1 cycle) is shown inthe following table.
Female Rat Population after 3 Months Age 0–3 3–6 6–9 9–12 12–15 15–18
Number 16.6 9.0 8.1 11.7 4.0 0
The sum of the number of female rats in each age group results in atotal population of female rats equal to 16.6 + 9.0 + 8.1 + 11.7 + 4.0 + 0, or49.4. So, after 3 months (1 cycle) the female rat population has grownfrom 42 to approximately 50.
Notice that in the table the number of female rats in each age groupis not rounded to the nearest integer. This is because when the values areto be used for further analysis, rounding off can mean a significantdifference in calculations over time even though it makes no sense to havea fractional part of a rat.
Copyright (c) 1995 by Thaves. Distributed from www.thecomics.com.
152 Chapter 3 • Matrix Operations and Applications
Exercises1. Use the preceding distribution table (Female Rat Population after 3
Months on page 151) and the process introduced in this lesson tocompute the following.
a. Calculate the number of newborn rats (aged 0–3) after 6 months(2 cycles).
b. Calculate the number of rats that survive in each age groupafter 6 months and move up to the next age group.
c. Use the results to parts a and b to show the distribution of therat population after 6 months. Approximately how many ratswill there be after 6 months?
d. Use your population distribution from part c to calculate thenumber of rats and the approximate number in each age groupafter 9 months (3 cycles). Continue this process to find thenumber of rats after 12 months (4 cycles).
e. Compare the original number of rats with the number of ratsafter 3, 6, 9, and 12 months. What do you observe?
f. What do you think might happen to this population if youextended the calculations to 15, 18, 21, . . . months?
For Exercises 2 and 3 use the following birth and survival rates for acertain species of deer.
153Lesson 3.4 • Population Growth: The Leslie Model, Part 1
b. Calculate the number of deer that survive in each age groupafter 2 years and move up to the next age group.
c. Arrange the initial population distribution in a row matrix andthe birth rates in a column matrix. Multiply the row matrixtimes the column matrix. Interpret this result.
3. Explore the possibility of multiplying the initial populationdistribution in a row matrix times some column matrix to find thenumber of deer after 2 years that move from:
a. The 0–2 group to the 2–4 group. (Hint: the column matrix thatyou use will need to contain several zeros in order to produce thedesired product.)
b. The 2–4 to the 4–6 group.
c. The 4–6 group to the 6–8 group.
d. The 6–8 group to the 8–10 group.
e. The 8–10 group to the 10–12 group.
4. Use the birth and survival rate information for Rattus norvegicusfrom this lesson (see the table on page 150) to find the populationtotal and distribution after 3 months (1 cycle) for the followinginitial populations.
a. [35 0 0 0 0 0]
b. [5 5 5 5 5 5]
5. Assume an initial deer population of [25 0 0 0 0 0]. Use the birthand survival rate information for the deer population in Exercises 2and 3 to find the population total and distribution after each of thefollowing time spans.
In Lesson 3.4, you found that it was possible to use an initial populationdistribution along with birth and survival rates to predict populationnumbers at future times. As you explored your model, you found that youcould look 2, 3, or even 4 cycles into the future. However, the arithmeticsoon became cumbersome. What do wildlife managers and urbanplanners do if they want to look 10, 20, or even more cycles into the future?
In Lesson 3.4 (Exercises 2 and 3), you began to get a glimpse of themodel that Leslie proposed. The use of matrices seems to hold the key. Andwith the aid of computer software or a calculator, looking ahead manycycles is not difficult. In fact, some very fascinating results are produced.
Return to the original rat model. If you multiply the originalpopulation distribution (P0) times a matrix that we will call L, you cancalculate the population distribution at the end of cycle 1 (P1).
155Lesson 3.5 • Population Growth: The Leslie Model, Part 2
The matrix L is called the Leslie matrix. This matrix is formed byaugmenting or joining the column matrix containing the birth rates ofeach age group and a series of column matrices that contain the survivalrates. Notice that the survival-rate columns contain the survival rates asone entry and zeros everywhere else. The survival rates (of which there isone less than the actual number of survival rates since no animal survivesbeyond the 15–18 age group) lie along the super diagonal that isimmediately above the main diagonal of the matrix.
When the matrix L is multiplied by a population distribution Pk, anew population distribution Pk+1 results. To find population distributionsat the end of other cycles, the process can be continued.
P1 = P0L
P2 = P1L = (P0L)L = P0(LL) = P0L2
In general, Pk = P0Lk.
ExampleUse the formula Pk = P0L
k to find the population distribution for the ratsafter 24 months (8 cycles) and the total population of the rats.
Solution:
The population distribution after 8 cycles is
P8 = P0L8 = [15 9 13 5 0 0]
= [21.03 12.28 10.90 9.46 7.01 4.27]
Total population = 21.03 + 12.28 + 10.90 + 9.46 + 7.01 + 4.27 = 64.95, or approximately 65 rats.
156 Chapter 3 • Matrix Operations and Applications
ExercisesNote: For the following exercises, you need to have access to either agraphing calculator or computer software that performs matrixoperations.
1. Use the original population distribution, [15 9 13 5 0 0], and the Leslie matrix from theRattus norvegicus example to find the following.
a. The population distribution after 15 months (5 cycles)
b. The total population after 15 months (Hint:Multiply P0 L
5 times a column matrix consistingof six 1s.)
c. The population distribution and the totalpopulation after 21 months
2. Suppose the Rattus norvegicus start dying off from overcrowdingwhen the total female population for a colony reaches 250. Find howlong it will take for this to happen when the initial population is
a. [18 9 7 0 0 0].
b. [35 0 0 0 0 0].
c. [5 5 5 5 5 5].
d. [25 15 10 11 7 13].
3. a. Complete the table for the given cycles of Rattus norvegicus usingthe original population distribution of [15 9 13 5 0 0].
157Lesson 3.5 • Population Growth: The Leslie Model, Part 2
b. What do you observe about the growth rates?
c. Calculate the total populations for P25, P26, and P27. What is thegrowth rate between these successive years? Hint: To find thegrowth rate from P25 to P26, subtract the total population for P25from the total population for P26 and divide the result by thetotal population for P25.
4. One characteristic of the Leslie model is that growth does stabilizeat a rate called the long-term growth rate of the population. Asyou observed in Exercise 3, the growth rate of Rattus norvegicusconverges to about 3.04%. This means that for a large enough k,the total population in cycle k will equal about 1.0304 times thetotal population in the previous cycle.
a. Find the long-term growth rate of the total population for eachof the initial population distributions in Exercise 2.
b. How does the initial population distribution seem to affect thelong-term growth rate?
5. Again, consider the deer species from Lesson 3.4. The birth andsurvival rates follow.
158 Chapter 3 • Matrix Operations and Applications
In his study of the application of matrices to population growth, P. H.Leslie was particularly interested in the special case in which the birthrate matrix has only one nonzero element. The following example fallsinto this special case.
6. Suppose there is a certain kind of bug that lives at most 3 weeksand reproduces only in the third week of life. Fifty percent of thebugs born in one week survive into the second week, and 70% ofthe bugs that survive into their second week also survive into theirthird week. On the average, six new bugs are produced for eachbug that survives into its third week. A group of five 3-week-oldfemale bugs decide to make their home in a storage box in yourbasement.
a. Construct the Leslie matrix for this bug.
b. What is P0?
c. How long will it be before there are at least 1,000 female bugsliving in your basement?
7. Exercise 6 is an example of a population that grows in waves. Willthe population growth for this population stabilize in any way overthe long run? To explore this question, make a table of thepopulation distributions P22 through P30.
a. Examine the population change from one cycle to the next. Canyou find a pattern in the population growth?
b. Examine the population change from P22 to P25, P23 to P26, P24 toP27, P25 to P28, P26 to P29, and P27 to P30. Are you surprised at theresults? Why?
8. a. Change the initial population in Exercise 6 to P0 = [4 4 4] andrepeat the instructions in Exercise 7 looking at the totalpopulation growth for each cycle.
b. Examine the changes in successive age groups from P22 to P25,P23 to P26, P24 to P27, P25 to P28, P26 to P29, and P27 to P30. Make aconjecture based on your results.
9. Using mathematical induction, prove that Pk = P0Lk for any original
population P0 and Leslie matrix L, where k is a natural number.
Modeling Projects10. Search the Web for applications of the Leslie matrix model in
The following application of the Leslie matrix to a population mesticsheep in New Zealand was originally published in 1967. (See G. Caughley,“Parameters for Seasonally Breeding Populations,” Ecology 48(1967):834-839. Anton and Rorres (1987) developed the problem in their text,Elementary Linear Algebra with Applications. More recent references can befound by searching the Web for Leslie matrix applications.
This application involves what is referred to as the harvesting of apopu lation. The term harvesting is defined as removal of the animalsfrom the population. This could entail slaughtering some of the animalsas is the case of wild deer or caribou herds that grow too large to besupported by their habitat. It could also mean selling or relocating someof the animals to start a new herd or colony as is the case of domesticherds or wild colonies of animals such as beaver. The ultimate goal is tofind a stable distribution from which the population growth can beharvested at regular intervals so that the population can be heldconstant.
This model begins with an initial population that undergoes agrowth period that is described by the Leslie matrix. At the end of thegrowth period, however, a certain percentage of each age group in thedistribution is harvested. This is usually done in such a way that theunharvested population has the same age distribution as the initialpopulation. A plan for harvesting the same percentage of each age groupon a regular basis so the population remains the same after eachharvesting is called a sustainable harvesting policy
160 Chapter 3 • Matrix Operations and Applications
To describe this situation using matrices, suppose a Leslie matrix anda population distribution vector for n age groups of females in apopulation have been defined. If hi, for i = 1, 2, 3, ..., n, represents thefraction offemales harvested for each of the age groups, an n × n diagonalmatrix H called the harvesting matrix can be formed as follows.
If all the hi’s are the same value, then the harvesting is calleduniform. From the definition of a sustainable harvesting policy, it followsthat
Using matrix notation, where P represents the distribution at thebeginning of the growth period, L represents the Leslie matrix, and Hrepresents the harvesting matrix, this translates to
PL – PLH = P.
This model was applied in 1967 to a species of domestic sheep in NewZealand which has a lifespan of 12 years. The sheep were divided into 12age groups with a growth period of 1 year. Birth and survival rates foreach age group of sheep were found using demographics, and thefollowing Leslie matrix was developed.
H
h
h
h
hn
0 0 . . . 0
0 0 . . . 0
0 0 . . . 0
. . . .
. . . .
. . . .0 0 0 . . .
1
2
3=
[ ]
− =
Age� distributionat� end� ofgrowth� period
harvestAge� distributionat� beginning� ofgrowth� period
If left to reproduce without a har vesting policy in place, the sheepwould eventually approach a stable growthrate. If the shepard allows thisto hap pen without a harvesting policy, the income from selling the woolwould not cover the cost of feeding the flock. The stable growth rate thatcan be found using the Leslie matrix can be used to approximate auniform harvesting policy. In this case the uniform harvest ing policy isone in which roughly 18% of the sheep from each of the 12 age groups isharvested each year.
1. Write a summary of what you think are the important points of this chapter.
For Exercises 2 and 3, use the following matrices.
A = B = C = D =
2. a. How many elements does matrix B have?
b. What is the value of C12? Of C21?
3. Find the value of each of the given expressions. If it is not possible,state the reason why.
a. A + C
b. C – B
c. (A + C) – D
d. 2A + D
4. Your math club is planning a Saturday practice session for anupcoming math contest. For lunch the students ordered 35 Mexicanlunches, 6 bags of corn chips, 6 containers of salsa, and 12 six-packs of cold drinks.
a. Write this information in a row matrix L. Label your matrix.
b. Interpret L2 and L4.
c. Suppose that the club pays $4.50 per lunch, $1.97 per bag ofcorn chips, $2.10 for each container of salsa, and $2.89 for each six-pack of cold drinks. Use multiplication of a row and columnmatrix to find the total cost. Label your matrices.
5. A group of students is planning a retreat. They have contactedthree lodges in the vicinity to inquire about rates. They found thatCrystal Lodge charges $13.00 per person per day for lodging,$20.00 per day for food, and $5.00 per person for use of therecreational facilities. Springs Lodge charges $12.50 for lodging,$19.50 for meals, and $7.50 for use of the recreational facilities.Bear Lodge charges $20.00 per night for lodging, $18.00 a day formeals, and there is no extra charge for using the recreationalfacilities. Beaver Lodge charges a flat rate of $40.00 a day forlodging (meals included) and no additional fee for use of therecreational facilities.
a. Display this information in a matrix C. Label the rows andcolumns.
b. State the values of C22 and C43.
c. Interpret C13 and C31.
6. Mr. Jones has been shopping for a vacuum-powered cleaningsystem. He found one at Z-Mart and another model at BaseHardware. The Z-Mart system cost $39.50, disposal cartridges were6 for $24.50, and storage cases were $8.50 each. At Base Hardwarethe system cost $49.90, cartridges were 6 for $29.95, and cases were$12.50 each.
a. Write and label a matrix showing the prices for the three itemsat the two stores.
b. Mr. Jones decided to wait and see if the prices for the systemswould be reduced during the upcoming sales. When he wentback during the sales, the Z-Mart prices were reduced by 10%and the Base Hardware prices were reduced by 20%. Construct amatrix showing the sale prices for each of the three items at thetwo stores.
c. Use matrix subtraction to compute how much Mr. Jones couldsave for each item at the two stores.
d. Suppose Mr. Jones is interested in purchasing the systems forhimself and three of his friends. Use multiplication of a matrixby a scalar to find how much he would pay for each of the threeitems at the two stores at the sale prices.
164 Chapter 3 • Matrix Operations and Applications
7. The dimensions of matrices P, Q, R, and S are 3 × 2, 3 × 3, 4 × 3,and 2 × 3, respectively. If matrix multiplication is possible, find thedimensions of the following matrix products. If it is not possible,state why.
a. QP
b. RQ
c. QS
d. RPS
8. An artist creates plates and bowls from small pieces of coloredwoods. She currently has orders for five plates, three large bowls,and seven small bowls. Each plate requires 100 pieces of ebony, 800pieces of walnut, 600 pieces of rosewood, and 400 pieces of maple.It takes 200 ebony pieces, 1,200 walnut pieces, 1,000 rosewoodpieces, and 800 pieces of maple to make a large bowl. A small bowltakes 50 pieces of ebony, 500 walnut pieces, 450 rosewood pieces,and 400 pieces of maple.
a. Write a row matrix showing the current orders for this artist’swork.
b. Construct a matrix showing the number of pieces of wood usedin an individual plate or bowl.
c. Use matrix multiplication to compute the number of pieces ofeach type of wood the artist will need for the plates and bowlsthat are on order.
d. Suppose it takes the artist 3 weeks to fashion a plate, 4 weeks tomake a large bowl, and 2 weeks to complete a small bowl. Usematrix multiplication to show how long it will take the artist tofill all the orders for plates and bowls.
9. Tonya has money invested in three sports complexes in Smith City.Her return (annual) from a $50,000 investment in a tennis club is8.2%. She receives 6.5% from a $100,000 investment in a golf cluband 7.5% on a $75,000 investment in a soccer club. Use matrixmultiplication to find Tonya’s income from her investments for oneyear. Label your matrices.
10. Three music classes at Central High are selling candy as afundraiser. The number of each kind of candy sold by each of thethree classes is shown in the following table.
The profit for each type of candy is sour balls, 30 cents; chocolatechews, 50 cents; almond bars, 25 cents; and mint patties, 35 cents.Use matrix multiplication to compute the profit made by each classon its candy sales.
11. Use the following matrices to find the value of each of the givenexpressions. If the expression is not defined, give the reason.
A = B = C = D = [2 1 0] E = [4 3]
a. AB
b. BA
c. CA
d. DA + E
12. Write the transpose (AT) of matrix A, where
A = .
13. Let matrix
M = .
a. Calculate M2, M3, and M4.
b. Predict the components of M5 and check your prediction.
c. Generalize to Mn, where n is a natural number.
d. Prove your conjecture in part c using mathematical induction.
166 Chapter 3 • Matrix Operations and Applications
14. Complete the following statement: If a square matrix A has aninverse A–1, then the product AA–1 = the ______ matrix I, where I is a_________.
15. Which of the following matrices are inverses of each other? Explainyour answers.
a. and
b. and
c. and
16. The students at Central High are planning to hire a band for theprom. Their choices are bands A, B, and C. They survey theSophomore, Junior, and Senior classes and find the followingpercentages of students (regardless of sex) prefer the bands,
.
The student population by class and sex is:
.
Use matrix multiplication to find:
a. The number of males and females who prefer each band.
b. The total number of students who prefer each band.
Suppose the initial female population for the herd is given by
P0 = [22 22 18 20 7 2].
a. What is the expected lifespan of this mammal?
b. Construct the Leslie matrix for this population.
c. Determine the long-term growth rate for the herd.
d. Suppose this mammal starts dying off from overcrowding whenthe total female population for the herd reaches 520. How longwill it take for this to happen?
BibliographyAnton, H., and C. Rorres. 2010. Elementary Linear Algebra: Applications
Version. 10th ed. New York, NY: John Wiley & Sons.
Cozzens, M. B., and R. D. Porter. 1987. Mathematics and Its Applications.Lexington, MA: D.C. Heath and Company.
Kemeny, J. G., J. N. Snell, and G. L. Thompson. 1974. Introduction toFinite Mathematics. Englewood Cliffs, NJ: Prentice Hall.
Leslie, P. H. 1945. “On the Uses of Matrices in Certain PopulationMathematics.” Biometrika 33:183-212.
Maurer, S. B., and A. Ralston. 2004. Discrete Algorithmic Mathematics.3rd ed. Wellesley, MA: A. K. Peters Ltd.
168 Chapter 3 • Matrix Operations and Applications
North Carolina School of Science and Mathematics. 1988. New Topicsfor Secondary School Mathematics: Matrices. Reston, VA: NationalCouncil of Teachers of Mathematics.
Ross, K. A., and C. R. B. Wright. 2003. Discrete Mathematics. 5th ed.Englewood Cliffs, NJ: Prentice Hall.
Tuchinsky, Philip M. 1986. Matrix Multiplication and DC Ladder Circuits.Lexington, MA: COMAP, Inc.
![PSOD Lecture 2. MathCAD – vectors and matrix Matrix operations Matrix operations –Multiply by constant –Matrix transpose [ctrl]+[1] –Inverse [^][-][1]](https://static.documents.pub/doc/80x56/5516e69655034603568b4753/psod-lecture-2-mathcad-vectors-and-matrix-matrix-operations-matrix-operations-multiply-by-constant-matrix-transpose-ctrl1-inverse-1.jpg)