HSN.VM.C.6Common CoreMathNumber and QuantityGrades 9-12
HSN.VM.C.6: Using Matrices to Represent and Manipulate Data
In plain English: HSN.VM.C.6 is an advanced (+) Common Core number and quantity standard that asks students to use matrices to store and work with data, such as inventory tables, payoff matrices for games and decisions, and incidence matrices that record which points in a network are connected. It is usually taught in Precalculus, before matrix operations are studied on their own.
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
Common Core State Standards for Mathematics · Domain: Vector and Matrix Quantities (VM) · Cluster: Perform operations on matrices and use matrices in applications. Also written as HSN-VM.C.6 or N-VM.6 · Official standard
Students learn to use a matrix as an organized record of data: each row stands for one category (a store, a player's choice, a city) and each column for another (a product, the opponent's choice, a destination). They work with three kinds of data matrices. A data matrix stores counts such as inventory or sales, and students manipulate it by adding matrices and by multiplying by a price vector to get totals. A payoff matrix lists what one player or decision maker gains for every combination of choices, and students read it to compare choices and find each choice's worst case. An incidence (adjacency) matrix records which points in a network are connected: the entry in row i and column j is 1 when there is a direct link from point i to point j and 0 when there is not.
Students learn to read row sums and column sums as counts of links, to tell a two-way network (symmetric matrix) from a one-way network, and to use the square of a network matrix to count routes of exactly two links. On this page, matrices are written row by row with semicolons between the rows. For example, [3 0 1; 2 4 5] has first row 3, 0, 1 and second row 2, 4, 5, and a column vector such as [2; 7] has 2 on top of 7.
Learning Objectives
By the end of this lesson, students will be able to:
Organize a data set as a matrix with labeled rows and columns and state its dimensions
Add data matrices and multiply a quantity matrix by a price vector to answer questions about totals
Build and read a payoff matrix, including negative payoffs, and compare choices by their worst case
Write the incidence matrix of a one-way or two-way network and interpret its row sums, column sums and symmetry
Use the square of a network matrix to count routes of exactly two links, and check the count by tracing the routes
Prior Knowledge Required
Students should already be comfortable with:
Reading and building two-way tables of data 8.SP.A.4
Matrix dimensions (rows × columns) and naming an entry by its row and column
Multiplying a matrix by a column vector, row by row HSN.VM.C.11
Show a list of text messages sent yesterday by four friends and ask students to record it so that anyone can answer questions at a glance.
Warm-Up Prompt
"Yesterday Ana texted Ben and Dee, Ben texted Cy, Cy texted Ana, and Dee texted Ana. Make a table with the senders down the side and the receivers across the top, and write 1 when a text was sent and 0 when it was not. Who sent the most texts? Who received the most?"
Collect tables and agree on one: with rows and columns in the order Ana, Ben, Cy, Dee, the table is the matrix [0 1 0 1; 0 0 1 0; 1 0 0 0; 1 0 0 0]. Ana's row has two 1s (she sent 2 texts), and Ana's column has two 1s (she received 2). Ask why the table is not the same when you flip it across its diagonal: Ben texted Cy, but Cy did not text Ben. Tell students that this table is already a matrix, and that the lesson is about what else matrices can record and what you can compute from them.
Direct Instruction20-25 minutes
Introduce the three kinds of data matrices, one worked example at a time. Before each one, ask: "What do the rows stand for, what do the columns stand for, and what does one entry mean?"
Data matrix: adding
A bike shop has two stores (rows: Downtown, Mall) and stocks helmets, locks and lights (columns). Stock at the start of the week is S = [12 8 15; 6 10 9], and a delivery adds D = [4 6 0; 10 2 5].
Equation: S + D = [16 14 15; 16 12 14]: add matching entries, so the Mall now has 16 helmets
Data matrix times a price vector
Helmets sell for $40, locks for $25 and lights for $18, so the price vector is p = [40; 25; 18]. Find the retail value of each store's stock after the delivery.
A club will sell hot cocoa, lemonade or both at a fall fair. Estimated profits in dollars for cold, mild and hot weather (columns) are cocoa [180 90 20], lemonade [40 110 200] and both [120 130 110].
Equation: Row minimums are 20, 40 and 110, so selling both has the best worst case ($110)
Incidence matrix of a network
Direct flights run both ways between Austin and Chicago, Austin and Denver, Boise and Denver, and Chicago and Denver (Diagram 1). Use the order A, B, C, D.
Equation: F = [0 0 1 1; 0 0 0 1; 1 0 0 1; 1 1 1 0], row sums 2, 1, 2, 3, so Denver has the most direct flights
Two-step routes from the square
Compute F² for the flight matrix to count trips of exactly two flights between each pair of cities.
Equation: F² = [2 1 1 1; 1 1 1 0; 1 1 2 1; 1 0 1 3]: one two-flight trip from Boise to Chicago (through Denver)
After the second example, stress that a price vector only works if its entries are in the same order as the columns of the quantity matrix. After the third, point out that a payoff matrix records one player's or one decision maker's gains; a negative entry would be a loss. After the fourth, use Diagram 1 to show two facts about a two-way network: the matrix is symmetric (the entry in row i, column j equals the entry in row j, column i), and the diagonal is 0 because no flight goes from a city to itself. For the fifth, explain why squaring counts routes: the entry in row i, column j of F² adds up, over every middle city k, (links from i to k) × (links from k to j). Check one entry by tracing: Austin to Austin in two flights is Austin-Chicago-Austin or Austin-Denver-Austin, so the entry is 2.
Guided Practice15 minutes
Use Diagram 2. Four volleyball teams played a round robin: the Hawks beat the Owls and the Jays, the Owls beat the Jays and the Wrens, the Jays beat the Wrens, and the Wrens beat the Hawks. Pairs build the dominance matrix T with the winner's row and the loser's column, in the order Hawks, Owls, Jays, Wrens (answer: [0 1 1 0; 0 0 1 1; 0 0 0 1; 1 0 0 0]). Ask:
What do the row sums mean? (Wins: 2, 2, 1, 1.) What do the column sums mean? (Losses: 1, 1, 2, 2.)
Why is T not symmetric? (If the Hawks beat the Owls, the Owls did not beat the Hawks.)
The Hawks and Owls are tied with 2 wins. One way to break the tie is to also count two-step wins, the teams beaten by a team you beat. Compute T + T² and add each row (answer: T² = [0 0 1 2; 1 0 0 1; 1 0 0 0; 0 1 1 0], and the row sums of T + T² are 5, 4, 2, 3). The Hawks rank first, and the Wrens pass the Jays because they beat the strong Hawks.
Listen for these errors: putting the loser in the row, writing a 1 on the diagonal, and adding the entries of T² without first adding T.
Independent Practice15 minutes
Students complete three short problems on their own.
A theater sells adult, student and child tickets (columns) on Friday and Saturday (rows): [120 85 40; 150 110 65]. Prices are $14, $9 and $6. Find each night's ticket revenue with one matrix product. (Answer: [2685; 3480], so $2,685 on Friday and $3,480 on Saturday.)
A payoff matrix for a game gives Player R's points: [1 -3 2; 0 2 -1], with rows for R's choices and columns for the opponent's choices. Which row has the better worst case? (Row minimums are -3 and -1, so row 2.)
Studio A has a video link to studios B and C, but B and C are not linked. Write the incidence matrix in the order A, B, C and square it. (Answer: [0 1 1; 1 0 0; 1 0 0], and its square is [2 0 0; 0 1 1; 0 1 1]: B can reach C in two links, through A.)
Closure5 minutes
Exit ticket: (1) Towns X, Y and Z have two-way roads X-Y and Y-Z. Write the incidence matrix in the order X, Y, Z. (Answer: [0 1 0; 1 0 1; 0 1 0].) (2) In a payoff matrix for Player R, what does an entry of -4 mean? (R loses 4 points for that pair of choices.) (3) Which matrix operation turns a quantity matrix and a price vector into totals, and what must be true about the order of the prices?
Differentiation Strategies
For Struggling Students
Always label the rows and columns of a matrix in the margin before writing any entries, and keep the labels while computing
For networks, start with the drawing: go through the points one at a time and list every point it links to directly before filling in its row
Provide a matrix calculator for squaring network matrices so that attention stays on what the entries mean
For Advanced Students
Ask students to explain why the diagonal entries of F² equal the row sums of F for any two-way network with no loops
Ask what F + F² counts, and use it to find which pairs of cities are connected by at most two flights
For the club's payoff matrix, ask which choice is best if cold, mild and hot weather are equally likely, and compare the answer with the worst-case choice (this goes beyond the standard into expected value)
Assessment Guidance
What to Look For
Check that students can say in words what one entry, one row sum and one column sum mean in context, not only compute them. For networks, look for a correct choice between a symmetric matrix (two-way links) and a non-symmetric one (one-way links), zeros on the diagonal, and a count from the squared matrix that students confirm by listing the routes. For data matrices, check that the prices or weights are in the same order as the columns before multiplying, and that totals carry the right units.
02
Classroom Activities
3 Activities
1
Trail Network Matrix
20 minGroups of 3
Each group gets a printed map of a park with five trail junctions, P1 to P5, and six two-way trails: P1-P2, P1-P3, P2-P3, P2-P4, P3-P5 and P4-P5. Groups build the incidence matrix, read it and use its square to plan short hikes.
Procedure
One student reads the map, one writes the 5 × 5 matrix, and one checks that it is symmetric with zeros on the diagonal. Expected matrix: [0 1 1 0 0; 1 0 1 1 0; 1 1 0 0 1; 0 1 0 0 1; 0 0 1 1 0]
Find the row sums (2, 3, 3, 2, 2) and name the junctions where the most trails meet (P2 and P3)
Use a calculator to square the matrix. Read how many hikes of exactly two trails go from P2 to P5 (2: P2-P3-P5 and P2-P4-P5) and from P4 to P2 (0)
Trace every route you counted on the map to confirm the matrix
Discussion Questions
P4 and P2 are next to each other on the map, yet the two-trail count is 0. Why is that not a contradiction?
Why is every diagonal entry of the squared matrix equal to that junction's row sum?
If the park closes trail P2-P3, which entries of the matrix change?
Modification for Distance Learning
Share the map as an image and have each group fill in a shared spreadsheet grid for the matrix. A free online matrix calculator can square it.
2
Two-Finger Payoff Game
20 minPairs
Pairs play a short game, record the results and then write the payoff matrix that summarizes the rules. The matrix lets them compare the choices without playing hundreds of rounds.
Rules
On a signal, both players show one or two fingers
If the total is even, Player R wins that many points from Player C. If the total is odd, Player C wins that many points from Player R
Play 12 rounds and record each result as Player R's gain, using a negative number for a loss
Build the Matrix
Write the payoff matrix for Player R with rows R shows 1, R shows 2 and columns C shows 1, C shows 2. Expected matrix: [2 -3; -3 4]
Find the worst case in each row (-3 and -3) and the best case (2 and 4)
Compare the matrix with your 12 recorded rounds: did every result match an entry?
Challenge Variation
Going further: suppose R shows one finger with probability p and C chooses at random. Ask pairs to find the value of p that makes R's expected gain the same whatever C does (p = 7/12, with an expected gain of -1/12 point per round) and to decide whether the game is fair. This connects to expected value in the statistics standards.
3
School Store Order
20 minGroups of 3-4
Groups act as the school store's managers. They turn three grade-level orders into a matrix, combine it with a price vector and answer the questions a real store would ask.
Data
Grade 9 orders 18 hoodies, 30 T-shirts and 12 caps; grade 10 orders 24, 26 and 9; grade 11 orders 15, 35 and 20
Prices: hoodie $32, T-shirt $12, cap $15
Procedure
Write the order matrix O with one row per grade: [18 30 12; 24 26 9; 15 35 20]
Multiply O by the price vector [32; 12; 15] to get each grade's bill: [1116; 1215; 1200]
Find the column sums to see how many of each item to order from the supplier (57 hoodies, 91 T-shirts, 41 caps)
Write one question of your own that the matrix can answer, and trade it with another group
Discussion Questions
What would go wrong if the price vector were written as [12; 32; 15]?
Why can you not multiply the price vector by O in the other order?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Two-Way Flight Network and Its Incidence Matrix
Each line is a pair of direct flights, one in each direction, so the matrix is symmetric. The entry in row i and column j is 1 when there is a direct flight from city i to city j. The row sums count each city's direct connections.
Diagram 2: A One-Way Network: Round-Robin Results
In a dominance matrix the winner is the row and the loser is the column. Because each game has one winner, the matrix is not symmetric, and the row sums give each team's wins.
04
Homework Assignment
~30 min
HSN.VM.C.6 Homework: Matrices for Data, Payoffs and Networks
Directions: Label the rows and columns of every matrix you write. Write matrices row by row with semicolons between rows, as in the lesson. Explain what each answer means in the context of the problem. A calculator may be used to square matrices in Problem 6.
Part 1: Data Matrices (Problems 1-2)
A community garden records its harvest in pounds. Rows are plots A and B, and columns are tomatoes, peppers and squash. June: [22 9 14; 17 12 20]. July: [31 15 18; 26 10 24]. (a) Find the matrix of the total harvest for the two months. (b) Find July minus June and explain what its negative entry means.
A bakery supplies three cafes. Rows are Cafe 1, Cafe 2 and Cafe 3, and columns are dozens of bagels, muffins and scones: [5 3 2; 8 4 0; 6 6 3]. A dozen bagels costs $9, a dozen muffins $15 and a dozen scones $18. (a) Use a matrix product to find each cafe's bill. (b) How many dozens of each item must the bakery make? (c) What is the total of all three bills?
Part 2: Payoff Matrices (Problems 3-4)
In rock-paper-scissors, the winner of a round gains 1 point and the loser loses 1 point; a tie scores 0. (a) Write the payoff matrix for the row player, with rows and columns in the order rock, paper, scissors. (b) What is the entry for the row player choosing rock while the column player chooses scissors? (c) Compare the entry in row i, column j with the entry in row j, column i. What does this say about the game?
A farmer can plant corn, soybeans or wheat (rows). Profits in thousands of dollars for low, normal and high rainfall (columns) are [-6 42 55; 8 35 40; 15 28 22]. (a) Which crop has the best worst case? (b) Which crop has the best best case? (c) If a long-range forecast strongly predicts normal rainfall, which crop would you choose, and why?
Part 3: Networks (Problems 5-6)
On a class social site, Ava follows Blake and Cruz, Blake follows Ava and Dani, Cruz follows Ava, Dani follows Ava, Blake and Eli, and Eli follows Dani. (a) Write the incidence matrix with the follower as the row, in the order Ava, Blake, Cruz, Dani, Eli. (b) Who has the most followers, and which sum did you use? (c) Who follows the most people? (d) How many pairs follow each other, and how can you see this in the matrix?
A city bus system has two-way routes between the Library and the Mall, the Library and the School, the Mall and the Terminal, the Park and the School, the Park and the Terminal, and the School and the Terminal. (a) Write the incidence matrix in the order Library, Mall, Park, School, Terminal. (b) Square it. (c) How many trips of exactly two rides go from the Library to the Terminal? List them. (d) The Library and the Mall are directly linked, but the entry for two-ride trips between them is 0. Explain why.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Building Matrices
Every matrix labeled, entries in the right rows and columns
One matrix mislabeled or one row and column swapped
Matrices missing or unlabeled
Computation
Sums, products and squares all correct
One or two arithmetic errors
Most computations incorrect
Payoffs and Decisions
Worst and best cases found and used to justify a choice
Cases found but choice not justified
Missing or incorrect
Interpretation
Every answer explained in context, including symmetry and route counts
Some answers explained
No interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. Matrices are written row by row, as in the lesson plan.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A school library records how many books each grade checked out, with one row for each of grades 9, 10, 11 and 12 and one column each for fiction, nonfiction and graphic novels. What are the dimensions of this matrix?
Answer: B
Dimensions are rows × columns: 4 grades give 4 rows and 3 book types give 3 columns, so the matrix is 4 × 3. Choice A swaps rows and columns. Choices C and D count categories or entries instead of rows and columns.
Question 2 of 20 · Multiple Choice
A pizza shop's sales matrix is [14 22 9; 11 25 13], with rows Monday and Tuesday and columns small, medium and large. What does the entry 13 mean?
Answer: C
13 is in row 2 (Tuesday) and column 3 (large). Choice B reads the right column but the wrong row. Choice A has the right row but the wrong column, a sign that the labels were not written down.
Question 3 of 20 · Multiple Choice
Four towns P, Q, R and S are joined by two-way roads. The incidence matrix, in the order P, Q, R, S, is [0 1 1 0; 1 0 1 1; 1 1 0 0; 0 1 0 0]. Which town has the most direct roads?
Answer: B
The row sums are 2, 3, 2 and 1, so Q has 3 direct roads, to P, R and S. Choices A and C each have 2. Choice D, S, has the fewest: its only road goes to Q.
Question 4 of 20 · Multiple Choice
Which matrix cannot be the incidence matrix of a network of two-way trails?
Answer: C
Two-way links make the matrix symmetric: the entry in row i, column j must equal the entry in row j, column i. In choice C, row 1, column 2 is 1 but row 2, column 1 is 0, so there is a link from point 1 to point 2 without one back. That describes a one-way network. Choices A, B and D are all symmetric with zeros on the diagonal.
Question 5 of 20 · Multiple Choice
In a two-player game, the payoff matrix for Rhea is [4 -2; -1 3]. Rows are Rhea's strategies and columns are Colin's. Rhea plays strategy 1 and Colin plays strategy 2. What happens?
Answer: D
Read row 1, column 2: the entry is -2, so Rhea loses 2 points (to Colin). Choice C reads row 2, column 1, swapping the roles of the players. Choice A reads row 1, column 1.
Question 6 of 20 · Multiple Choice
A shoe store has two locations. January sales are [40 25; 32 18] and February sales are [36 30; 29 22], with rows store 1 and store 2 and columns jackets and boots. How many boots did store 2 sell in the two months together?
Answer: B
Add the matrices: [76 55; 61 40]. Store 2's boots are in row 2, column 2: 18 + 22 = 40. Choice A is store 1's boots and choice C is store 2's jackets, both from reading the wrong entry. Choice D is February only.
Question 7 of 20 · Multiple Choice
A snack bar sells hot dogs, pretzels and drinks. Quantities sold are [60 45 110; 85 70 150], with rows Friday and Saturday, and prices are $4, $3 and $2. What is Saturday's revenue?
Answer: C
Multiply the Saturday row by the price vector [4; 3; 2]: 85(4) + 70(3) + 150(2) = 340 + 210 + 300 = $850. Choice A is Friday's revenue, choice B adds the quantities without prices, and choice D is both days together.
Question 8 of 20 · Multiple Choice
Four players play every other player once in chess. The matrix, with the winner as the row and players in the order Ari, Bea, Cal, Dev, is [0 1 0 0; 0 0 1 0; 1 0 0 0; 1 1 1 0]. Who won the most games?
Answer: D
Wins are row sums: 1, 1, 1 and 3, so Dev won all 3 of his games. Adding columns instead gives losses: Dev's column is all 0 because he never lost, which is why reading columns would point to the wrong player.
Question 9 of 20 · Multiple Choice
In a matrix that records who follows whom on a social site, the row is the follower and the column is the account being followed. What does the sum of column j count?
Answer: A
Column j has a 1 in every row whose person follows j, so the column sum counts j's followers. Choice B is the row sum. Choice C needs pairs of symmetric entries, not one column.
Question 10 of 20 · Multiple Choice
M is the incidence matrix of a two-way network. What does the entry in row i, column j of M² count?
Answer: A
Each entry of M² adds, over every middle point k, (links from i to k) × (links from k to j), so it counts two-link routes. Choice B describes M itself. Choice C is a distance, which M² does not give directly: two points can be joined by one link and still have 0 two-link routes.
Question 11 of 20 · Multiple Choice
Ferries run both ways between islands W and X, W and Y, X and Y, and Y and Z. The incidence matrix in the order W, X, Y, Z is [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0]. How many routes of exactly two ferry trips start and end at Y?
Answer: D
A two-trip route from Y back to Y goes out to a neighbor and returns: Y-W-Y, Y-X-Y and Y-Z-Y, so there are 3. The diagonal entry of the squared matrix equals Y's row sum, 1 + 1 + 0 + 1 = 3. Choice A assumes the diagonal stays 0 after squaring.
Question 12 of 20 · Multiple Choice
A club compares three fundraisers. Profits in dollars for sunny and rainy weather are car wash [450 50], bake sale [300 260] and raffle [220 220]. Which choice has the best worst case?
Answer: B
The worst cases (row minimums) are $50, $260 and $220, and the largest of these is $260 for the bake sale. Choice A has the best best case ($450) but the worst worst case. Choice C has the same profit in both columns, but $220 is still less than $260.
Question 13 of 20 · Multiple Choice
Why is every diagonal entry 0 in the incidence matrix of a network of direct flights between cities?
Answer: D
A diagonal entry records a link from a point to itself, and no flight does that. Choice A is true for two-way networks but does not force the diagonal to be 0: a symmetric matrix can have nonzero diagonal entries.
Question 14 of 20 · Multiple Choice
One-way streets run from intersection 1 to 2, from 2 to 3, from 3 to 1 and from 1 to 3. Which matrix records them, with the starting intersection as the row?
Answer: C
Row 1 has 1s in columns 2 and 3, row 2 has a 1 in column 3, and row 3 has a 1 in column 1. Choice A puts the destination in the row instead. Choice B treats every street as two-way. Choice D leaves out the street from 3 to 1.
Question 15 of 20 · Short Answer
Liam texts Mia and Noah, Mia texts Noah, Noah texts Olga, and Olga texts Liam and Mia. Write the matrix with the sender as the row, in the order Liam, Mia, Noah, Olga. Who sent the most texts, and who received the most?
[0 1 1 0; 0 0 1 0; 0 0 0 1; 1 1 0 0]. Row sums (texts sent) are 2, 1, 1, 2, so Liam and Olga each sent 2. Column sums (texts received) are 1, 2, 2, 1, so Mia and Noah each received 2.
Question 16 of 20 · Short Answer
Rosa and Chen each say "high" or "low" at the same moment. If both say high, Rosa wins 3 points. If both say low, Rosa wins 1 point. If they say different words, Chen wins 2 points. Write Rosa's payoff matrix with rows Rosa high, Rosa low and columns Chen high, Chen low, and give the worst case of each row.
[3 -2; -2 1]. When the words differ, Rosa loses 2 points, so those entries are -2. The worst case of each row is -2, so neither choice is safer; saying high has the better best case (3).
Question 17 of 20 · Short Answer
A charity 5K sells adult ($35), student ($20) and child ($10) registrations. Registrations are [140 85 60; 35 40 25], with rows online and race day. Use a matrix product to find the revenue from each kind of sale and the total.
[140 85 60; 35 40 25][35; 20; 10] = [7200; 2275]: $7,200 online (4900 + 1700 + 600) and $2,275 on race day (1225 + 800 + 250), for a total of $9,475.
Question 18 of 20 · Short Answer
A bike-share program counts rides. Rows are the North and South stations, and columns are weekday and weekend. Last year: [420 380; 510 300]. This year: [465 350; 540 330]. Find this year minus last year and explain the negative entry.
This year minus last year is [45 -30; 30 30]. The -30 is in row North, column weekend: weekend rides at the North station fell by 30. Every other count rose.
Question 19 of 20 · Short Answer
Four soccer teams played each other once: the Lions beat the Bears and the Wolves, the Bears beat the Foxes and the Wolves, the Foxes beat the Lions, and the Wolves beat the Foxes. Write the dominance matrix T (winner as row, order Lions, Bears, Foxes, Wolves). The Lions and Bears both have 2 wins; use the row sums of T + T² to break the tie.
T = [0 1 0 1; 0 0 1 1; 1 0 0 0; 0 0 1 0]. T² = [0 0 2 1; 1 0 1 0; 0 1 0 1; 1 0 0 0], so T + T² = [0 1 2 2; 1 0 2 1; 1 1 0 1; 1 0 1 0] with row sums 5, 4, 3, 2. The Lions rank first: the teams they beat won more games.
Question 20 of 20 · Short Answer
Four towns have two-way roads with incidence matrix N = [0 1 1 1; 1 0 0 1; 1 0 0 0; 1 1 0 0] in the order 1, 2, 3, 4. (a) Find N² (a calculator is allowed). (b) How many two-road routes go from town 2 to town 4? (c) What does the diagonal entry for town 1 in N² mean?
(a) N² = [3 1 0 1; 1 2 1 1; 0 1 1 1; 1 1 1 2]. (b) Row 2, column 4 is 1: the route 2-1-4. (c) The entry 3 counts routes that leave town 1 and return in two roads (to towns 2, 3 or 4 and back), which is the number of roads at town 1.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.VM.C.6 mean?
HSN.VM.C.6 asks students to use matrices to represent and manipulate data. The standard names two kinds of data on purpose: payoffs, as in the payoff matrix of a game or a decision, and incidence relationships in a network, as in a matrix that records which cities have direct flights. Students also use everyday data matrices such as inventory or sales, and they compute with them to answer questions: totals, changes, worst cases and numbers of routes.
Is HSN.VM.C.6 taught in Algebra 2 or Precalculus?
It is usually taught in Precalculus. The (+) in front of the standard marks it as additional mathematics for students who take advanced courses, so some Algebra II or discrete mathematics courses include it and others leave it out. It fits well at the start of a matrix unit, because data give students a reason to add and multiply matrices before those operations are studied on their own (HSN.VM.C.7 and HSN.VM.C.8).
What is an incidence matrix of a network?
It is a square matrix with one row and one column for each point (vertex) of the network. The entry in row i and column j is 1 if there is a direct link from point i to point j and 0 otherwise. Many textbooks call this an adjacency matrix. For a two-way network, the matrix is symmetric; for a one-way network, such as follows on a social site or wins in a tournament, it usually is not.
What is a payoff matrix?
A payoff matrix lists what one player gains for every combination of choices. The rows are that player's choices, the columns are the other player's choices (or the possible conditions, such as the weather), and each entry is a gain. A negative entry is a loss. For example, [2 -3; -3 4] gives Player R's points in a finger game: R wins 2 when both show one finger and loses 3 when the fingers differ.
Why use a matrix instead of an ordinary table?
A matrix is a table that you can compute with. Once the data are in a matrix, one addition combines two months of sales, one product with a price vector gives every store's revenue at once, and one squaring counts every two-step route in a network. These calculations scale to large data sets with technology, which is how matrices are used in practice.
What does squaring a network matrix tell you?
The entry in row i and column j of M² is the number of routes from point i to point j that use exactly two links. The reason is the row-by-column rule: the entry adds, over every possible middle point k, the product (links from i to k) × (links from k to j), and that product is 1 exactly when i-k-j is a route. The diagonal entries of M² for a two-way network equal the number of links at each point.
What are common mistakes when building a matrix from data?
A common error is swapping rows and columns, for example putting the loser in the row of a tournament matrix or the destination in the row of a one-way network. Others are writing a symmetric matrix for a one-way network, putting 1s on the diagonal of a network matrix, and multiplying by a price vector whose entries are in a different order from the columns. Labeling every row and column before writing entries prevents most of these.
Do students need to compute large matrix products by hand?
No. The standard is about representing data and interpreting the results. Students should compute small cases by hand, such as a 2 × 3 matrix times a price vector, so that they understand what the answer means, and they can use a graphing calculator or a matrix app for larger products and for squaring a 5 × 5 network matrix.
How can students check that a network matrix is right?
For a two-way network, check that the matrix is symmetric and that the diagonal is 0. Then compare each row sum with the drawing: it should equal the number of links at that point. For counts from M², pick one entry and list the routes by hand; if the list and the entry agree, the matrix is very likely correct.
How does HSN.VM.C.6 connect to other standards?
It is the application standard of the matrix cluster. The operations it uses are developed in HSN.VM.C.7 (scalar multiples, such as doubling every payoff), HSN.VM.C.8 (adding and multiplying matrices) and HSN.VM.C.9 (properties of multiplication). Payoff matrices lead to expected payoffs in HSS.MD.B.5, and network matrices appear again in discrete mathematics and computer science courses.
07
Related Standards
5 standards
These standards connect to HSN.VM.C.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.SP.A.4Prerequisite
Display and interpret bivariate categorical data in a two-way table
