HSN.VM.C.7Common CoreMathNumber and QuantityGrades 9-12
HSN.VM.C.7: Multiplying Matrices by Scalars
In plain English: HSN.VM.C.7 is an advanced (+) Common Core number and quantity standard, usually taught in Precalculus. Students multiply a matrix by a scalar, a single real number, by multiplying every entry by that number, and they use scalar multiples to rescale data such as game payoffs, prices and unit conversions. The result always has the same dimensions as the original matrix.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
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.7 or N-VM.7 · Official standard
Students learn scalar multiplication: to multiply a matrix A by a real number k, multiply every entry of A by k. The new matrix kA has the same number of rows and columns as A, and entry (i, j) of kA is k times entry (i, j) of A. The standard names a typical use: when all of the payoffs in a game are doubled, the new payoff matrix is 2 times the old one. Students also use scalars to raise or cut prices by a percent, to convert units, and to scale a recipe.
Students work with positive, negative, fractional and decimal scalars, find an unknown scalar or entry from a given product, and decide whether one matrix is a scalar multiple of another. On this page, matrices are written row by row, with rows separated by semicolons: [3 -1; -2 4] has first row 3, -1 and second row -2, 4, and [5; -2; 1] is a single column.
Learning Objectives
By the end of this lesson, students will be able to:
Multiply a matrix of any dimensions by a positive, negative, fractional or decimal scalar
Explain why kA has the same dimensions as A and describe entry (i, j) of kA
Find an unknown scalar or unknown entries from an equation kA = B
Model rescaling situations, such as doubled game payoffs, percent price changes and unit conversions, with a scalar multiple and interpret the entries
Prior Knowledge Required
Students should already be comfortable with:
Multiplying positive and negative fractions and decimals 7.NS.A.2
Percent increase and decrease 7.RP.A.3
Using a matrix to organize data, such as a payoff table HSN.VM.C.6
Naming matrix dimensions as rows × columns and locating entry (i, j)
Post a small price table and ask students to rewrite it after a change that affects every value:
Warm-Up Prompt
"A movie theater charges these ticket prices in dollars. Matinee: adult 8, child 5. Evening: adult 12, child 7. For a holiday, the theater cuts every price in half. Write the new table. A friend says that cutting the table in half means halving only the first row. How would you answer?"
The new prices are matinee 4 and 2.50, evening 6 and 3.50. Write both tables as matrices, [8 5; 12 7] and [4 2.5; 6 3.5], and point out that the change applied to every entry, not only one row. The friend's version would leave evening prices unchanged, which is not what the theater announced. Tell students that multiplying every entry of a matrix by the same number is an operation with a name: scalar multiplication.
Direct Instruction15 minutes
Define a scalar as a single real number, as opposed to a matrix. Use Diagram 1 to show the rule, then give the procedure:
Note the dimensions of A: an m × n matrix has m rows and n columns. The answer kA will also be m × n.
Multiply each entry by k: entry (i, j) of kA is k times entry (i, j) of A. Go row by row so no entry is skipped.
Watch the signs: a negative scalar changes the sign of every nonzero entry. A negative entry times a negative scalar gives a positive entry.
Simplify each entry: reduce fractions and, in money contexts, round to cents only at the end.
Interpret: in a context, say what one entry of kA means, including its units.
Work through the examples below. Before each, ask students to predict the dimensions of the answer.
Doubling game payoffs
In a two-player game, the row player's payoffs in points are P = [3 -1; -2 4]. For the final round, all payoffs are doubled.
Equation: 2P = [6 -2; -4 8]
Percent increase
Drink prices in dollars at two stores (rows) for small, medium and large (columns) are [2.50 3.00 3.50; 2.25 2.75 3.25]. Every price rises 20%, so multiply by 1.2.
Equation: 1.2M = [3.00 3.60 4.20; 2.70 3.30 3.90]
Negative fractional scalar
Compute -1/2 times the 3 × 2 matrix [4 -6; 0 10; -8 2].
Equation: [-2 3; 0 -5; 4 -1], still 3 × 2
Finding the scalar
Find k if k[4 -2; 6 10] = [-10 5; -15 -25]. Divide any nonzero entry of the result by the matching entry: -10/4 = -5/2, and every other pair gives the same ratio.
Equation: k = -5/2
Unit conversion
A bakery records sales in dozens: rows are Saturday and Sunday, columns are bagels and muffins, D = [4 6; 3 5]. Convert to single items.
Equation: 12D = [48 72; 36 60]
After the examples, name three facts students can check with any matrix: 1A = A, 0A is a zero matrix of the same size, and (-1)A, written -A, replaces each entry with its opposite. Also show that scaling twice is the same as scaling once by the product: 3(2P) = 6P, because every entry is multiplied by 2 and then by 3. Use Diagram 2 to show a geometric meaning: when the columns of a matrix are the vertices of a figure, multiplying by 2 doubles every coordinate.
Guided Practice10-15 minutes
Pairs work five items, one at a time, and compare after each: (a) 3[2 -1 0; 4 5 -3] (answer: [6 -3 0; 12 15 -9]); (b) -4[1/2 3; -2 0] (answer: [-2 -12; 8 0]); (c) 0.25[8 12; -20 6] (answer: [2 3; -5 1.5]); (d) find k if k[6 -9] = [-4 6] (answer: k = -2/3); (e) find a and b if 3[a 2; -1 b] = [12 6; -3 -15] (answer: a = 4, b = -5). Listen for these errors: multiplying only the first row or column, adding k to each entry instead of multiplying, losing a sign with a negative scalar, and dividing the wrong way when finding k.
Independent Practice10 minutes
Students work alone on five items: (1) -2[3 0; -7 4] (answer: [-6 0; 14 -8]); (2) (1/3)[9 -6 12; 3 0 -15] (answer: [3 -2 4; 1 0 -5]); (3) a card game's payoff matrix [2 -3; -1 5] is tripled for a bonus round (answer: [6 -9; -3 15]); (4) a matrix of practice hours [1.5 2; 0.75 3] is converted to minutes (answer: 60 times the matrix, [90 120; 45 180]); (5) find k if k[5 -10; 2.5 0] = [-2 4; -1 0] (answer: k = -2/5). For items 3 and 4, students write one sentence explaining what the entry in row 2, column 1 means.
Closure5 minutes
Exit ticket: (1) Compute -3[2 -4; 1 0]. (Answer: [-6 12; -3 0].) (2) Is [6 9; 3 12] a scalar multiple of [2 3; 1 4]? If so, give the scalar. (Yes, k = 3.) (3) If A is a 2 × 3 matrix, what are the dimensions of 5A, and why? (2 × 3, because scalar multiplication changes the entries, not the number of rows or columns.)
Differentiation Strategies
For Struggling Students
Have students rewrite kA as a matrix of products first, such as [3(2) 3(-1); ...], before simplifying any entry
Start with whole-number scalars and a 2 × 2 matrix, then move to fractions, decimals and negative scalars
For "find k" items, have students divide one pair of matching entries and then test that k on every other entry
For Advanced Students
Ask students to explain why [2 0; 1 3] cannot be a scalar multiple of [4 1; 2 6], using only the entries in row 1, column 2
Ask students to prove that c(dA) = (cd)A for any 2 × 2 matrix A by writing A with letter entries
Ask how a price matrix changes after a 10% increase followed by a 10% decrease, and explain with one scalar why the prices do not return to the start
Assessment Guidance
What to Look For
Check that students multiply every entry, including zeros and entries in the last row, and that the dimensions of the answer match the original matrix. With negative scalars, look for correct signs on every entry. In context problems, students should choose the scalar from the situation (1.2 for a 20% increase, 12 for dozens to items, 2 for doubled payoffs) and explain what one entry of the new matrix means with units. When finding k, students should confirm that the same k works for every entry, not only the first one.
02
Classroom Activities
3 Activities
1
Raise the Stakes: Scaling Game Payoffs
15 minPairs
Pairs study a payoff matrix for a two-player game and rescale it for different rounds of a tournament, the situation the standard names. They then ask whether the rescaled game changes which strategy is best.
Setup
Player R chooses a row (Strategy 1 or 2) and Player C chooses a column (X, Y or Z). The entries are the points Player R wins; a negative entry means Player R loses points to Player C
Payoff matrix: G = [4 -2 1; -3 5 0]
Procedure
Write the payoff matrix for a championship round where payoffs are doubled (answer: 2G = [8 -4 2; -6 10 0])
Write the matrix for a practice round where payoffs are cut in half (answer: [2 -1 0.5; -1.5 2.5 0])
Write the matrix for a round worth one and a half times as much (answer: [6 -3 1.5; -4.5 7.5 0])
Write -G and explain what it represents if every point Player R wins is a point Player C loses (answer: [-4 2 -1; 3 -5 0], the payoffs from Player C's side)
Discussion Questions
If Player C always picks column X, which strategy should Player R choose in the original game? Does the answer change in the doubled or halved rounds?
Why does a positive scalar keep the order of the payoffs in each column, while -1 reverses it?
Which entry of G stays the same in every rescaled round, and why?
2
Scalar Match Card Sort
15 minPairs
Pairs receive 14 cards: 4 matrix cards, 4 scalar cards and 6 result cards. They build four matching sets of matrix, scalar and result. Two result cards are decoys built from common errors, and pairs must explain what went wrong on each.
Sort result cards by dimensions first, and match each to a matrix card with the same dimensions
Find the scalar for each pair by dividing one pair of matching entries, then check every other entry
Set aside the two result cards that match no scalar. For [5 2; 7 4], the error is adding 4 to each entry of [1 -2; 3 0]; for [4 -8; 3 0], only the first row was multiplied
Challenge Variation
Each pair writes two new decoy cards for a classmate's matrix, one from a sign error with a negative scalar and one from a skipped entry, and trades them with another pair to diagnose.
3
Smoothie Shop Recipe Scaling
15 minGroups of 3
Groups use a recipe matrix to plan orders of different sizes and to convert units, choosing the scalar from the situation each time.
The Recipe Matrix
One batch makes 4 servings. Rows: Berry, Mango, Green. Columns: cups of fruit, cups of yogurt, cups of juice
R = [2 1 1.5; 1.5 1 2; 1 0.5 2.5]
Procedure
Scale R for 10 servings (scalar 10/4 = 2.5; answer: [5 2.5 3.75; 3.75 2.5 5; 2.5 1.25 6.25])
Scale R for 2 servings (scalar 0.5; answer: [1 0.5 0.75; 0.75 0.5 1; 0.5 0.25 1.25])
Convert R from cups to fluid ounces, with 8 fluid ounces in a cup (answer: [16 8 12; 12 8 16; 8 4 20])
Find the matrix for 10 servings in fluid ounces two ways: scale then convert, and convert then scale. Both give 20R
Discussion Questions
Why is the scalar for 10 servings 2.5 and not 10?
What single scalar turns R into amounts in fluid ounces for 10 servings, and why does the order of the two steps not matter?
Fruit is sold by the pound, not by the cup. Could one scalar convert the whole matrix to pounds? What would you need to know?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Scalar Multiplication Entry by Entry
To find 3A, multiply each of the six entries of the 2 × 3 matrix A by 3. The result has the same dimensions as A, and each entry sits in the same position as the entry it came from.
Diagram 2: A Scalar Multiple of a Coordinate Matrix
Drawn to scale. The columns of T are the vertices (1, 1), (4, 1) and (2, 3) of a triangle. The matrix 2T has columns (2, 2), (8, 2) and (4, 6), so every coordinate doubles and the larger triangle has sides twice as long.
04
Homework Assignment
~30 min
HSN.VM.C.7 Homework: Multiplying Matrices by Scalars
Directions: Show the product for each entry before simplifying. State the dimensions of every matrix you write. In Part 2, say what the scalar is and why it fits the situation, and give units for your answers.
Part 1: Computing and Reasoning with Scalar Multiples (Problems 1-3)
(a) Find k if k[-3 6; 1.5 9] = [4 -8; -2 -12]. (b) Find x and y if 4[x 3; -2 y] = [10 12; -8 -6]. Show how you checked each answer.
Let A = [2 -5; 4 1]. Decide whether each matrix B is a scalar multiple of A. If it is, give the scalar k with B = kA; if not, explain why no scalar works. (a) B = [-6 15; -12 -3] (b) B = [1 -2.5; 2 0.5] (c) B = [4 -10; 8 -2]
Part 2: Scalar Multiples in Context (Problems 4-6)
In a two-player game, Player 1 picks a row and Player 2 picks a column. The points Player 1 wins are given by [5 -2 0; -4 3 1]. (a) For the final round all payoffs are tripled. Write the new payoff matrix. (b) Every point Player 1 wins is lost by Player 2. Write the matrix of Player 2's payoffs for the original game as a scalar multiple, and explain your scalar.
A bike shop's hourly rental rates in dollars have rows city bike, mountain bike, e-bike and columns weekday, weekend: [8 10; 12 15; 20 25]. (a) The shop raises every rate by 15%. Write the new rate matrix. (b) In summer, the shop takes 20% off the new rates. Write the summer rate matrix. (c) Show that one scalar turns the original matrix into the summer matrix, and explain why summer prices are not 5% below the original.
A garden center records the cubic yards of mulch two landscapers ordered in March, April and May: [3 4.5 2; 6 1.5 5] (rows: landscapers; columns: months). Mulch costs $36 per cubic yard. Write the cost matrix as a scalar multiple, compute it, and explain what the entry in row 2, column 3 means.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Computation
Every entry multiplied correctly, signs and fractions right
One or two entry errors
Several entries wrong or skipped
Dimensions
Dimensions stated and unchanged in every answer
Dimensions stated for some answers
Dimensions missing or changed
Finding k and Unknown Entries
Scalar or entries found and checked on every entry
Found but not checked, or one error
Missing or incorrect
Context
Scalar justified and entries interpreted with units
Correct matrix, interpretation incomplete
Scalar does not fit the situation
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, with rows separated by semicolons.
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
Compute 4[3 -2; 0 5].
Answer: A
Multiply every entry by 4: 4(3) = 12, 4(-2) = -8, 4(0) = 0, 4(5) = 20. Choice B adds 4 to each entry instead of multiplying. Choice C multiplies only the first entry, and choice D multiplies only the first row.
Question 2 of 20 · Multiple Choice
Compute -2[-1 4; 3 0].
Answer: B
-2(-1) = 2, -2(4) = -8, -2(3) = -6 and -2(0) = 0. Choice A keeps the sign of the 4, a sign error on one entry. Choice C multiplies by +2 instead of -2. Choice D adds -2 to each entry.
Question 3 of 20 · Multiple Choice
A is a 4 × 3 matrix. How many entries does 5A have?
Answer: C
Scalar multiplication keeps the dimensions, so 5A is also 4 × 3 and has 4 · 3 = 12 entries. Only the values of the entries change. Choice A multiplies the number of entries by 5. Choices B and D multiply 5 by the number of rows (20) or by the number of columns (15) instead of counting rows times columns.
Question 4 of 20 · Multiple Choice
Compute (1/2)[6 -9; 4 1].
Answer: D
Half of each entry: 3, -4.5, 2 and 0.5. Choice A halves only the first column. Choice B multiplies by 2 instead of 1/2. Choice C adds 1/2 to each entry.
Question 5 of 20 · Multiple Choice
In a game, the row player's payoffs are [-1 3 2; 0 -4 5]. For a special round, all of the payoffs are doubled. Which matrix gives the new payoffs?
Answer: A
Doubling every payoff means multiplying the matrix by 2, so each entry doubles and each sign stays the same. Choice B adds 2 to each payoff. Choice C doubles only the first row. Choice D multiplies by -2, which would also reverse who wins.
Question 6 of 20 · Multiple Choice
Find k if k[2 -8; 5 1] = [-6 24; -15 -3].
Answer: B
Divide matching entries: -6/2 = -3, 24/(-8) = -3, -15/5 = -3 and -3/1 = -3. Every entry gives the same ratio, so k = -3. Choice A drops the sign. Choice C divides the wrong way (2 divided by -6). Choice D subtracts 2 from -6 instead of dividing.
Question 7 of 20 · Multiple Choice
Find a and b if 5[a -1; 2 b] = [15 -5; 10 -20].
Answer: C
Entry by entry, 5a = 15 and 5b = -20, so a = 3 and b = -4. Choice A multiplies by 5 instead of dividing. Choice B subtracts 5 from the result entries. Choice D has both signs wrong.
Question 8 of 20 · Multiple Choice
Which matrix is a scalar multiple of [1 -2; 3 4]?
Answer: D
Choice D is -3 times the matrix: -3(1) = -3, -3(-2) = 6, -3(3) = -9, -3(4) = -12. In choice A the last entry was not doubled, so no single scalar works. Choice B scales only the first column. Choice C adds 2 to each entry, which is not a scalar multiple.
Question 9 of 20 · Multiple Choice
A price matrix P lists prices in dollars. Every price increases by 25%. Which expression gives the matrix of new prices?
Answer: C
A 25% increase keeps 100% of each price and adds 25% more, so each entry becomes 1.25 times the old price. Choice A gives only the increase, not the new price. Choice B adds 25 cents, and adding a number to a matrix is not defined anyway. Choice D treats 25% as 25.
Question 10 of 20 · Multiple Choice
A matrix lists ticket prices in dollars. Which scalar multiple gives the same prices in cents?
Answer: B
There are 100 cents in a dollar, so multiply every entry by 100; for example, $2.35 becomes 235 cents. Choice A converts cents to dollars, the opposite direction. Choices C and D use the wrong number of cents per dollar.
Question 11 of 20 · Multiple Choice
In a zero-sum game, every point Player 1 wins is lost by Player 2. Player 1's payoffs are [5 -8; 0 2]. Which matrix gives Player 2's payoffs?
Answer: B
Player 2 gets the opposite of each payoff, so the matrix is (-1) times Player 1's matrix: [-5 8; 0 -2]. A win of 5 for Player 1 is a loss of 5 for Player 2, and a loss of 8 is a win of 8. Choice A makes every entry positive, which confuses the opposite with absolute value. Choice C swaps the rows, and choice D changes the sign of only the positive entries.
Question 12 of 20 · Multiple Choice
A matrix A is multiplied by 4, and the result is multiplied by -1/2. Which single scalar gives the same final matrix?
Answer: D
Each entry is multiplied by 4 and then by -1/2, so it is multiplied by 4 · (-1/2) = -2 in all. Choice A adds the scalars instead of multiplying them. Choice B divides 4 by -1/2. Choice C subtracts -1/2 from 4.
Question 13 of 20 · Multiple Choice
A carpenter's matrix of board lengths in feet is F = [4 6; 10 3]. Which matrix gives the lengths in inches?
Answer: A
There are 12 inches in a foot, so multiply every entry by 12: 12F = [48 72; 120 36]. Choice B adds 12 instead of multiplying. Choice C divides by 12, which converts in the wrong direction. Choice D converts only the first entry.
Question 14 of 20 · Multiple Choice
A student computes -3[2 -5; -1 4] and writes [-6 -15; -3 -12]. What is the correct product?
Answer: C
Multiply every entry by -3: -3(2) = -6, -3(-5) = 15, -3(-1) = 3 and -3(4) = -12. The student gave the two negative entries negative products, but a negative times a negative is positive. Choice A multiplies by 3 instead of -3. Choice B adds -3 to each entry. Choice D multiplies only the first row.
Question 15 of 20 · Short Answer
Compute -1/5 [10 -25 0; -5 15 30] and state the dimensions of the result.
Multiply every entry by -1/5: [-2 5 0; 1 -3 -6], a 2 × 3 matrix, the same dimensions as the original. Check the signs: a negative entry times -1/5 becomes positive, so -25 becomes 5 and -5 becomes 1.
Question 16 of 20 · Short Answer
Is [8 -12; 6 20] a scalar multiple of [2 -3; 1.5 4]? Explain.
No. Divide matching entries: 8/2 = 4, -12/(-3) = 4 and 6/1.5 = 4, but 20/4 = 5. A scalar multiple needs one scalar for every entry, so no k works. A common error is to check only the first entry or only the first row.
Question 17 of 20 · Short Answer
A card game's payoff matrix, in points for the row player, is [6 -4; -2 3]. In a warm-up round, all payoffs are cut in half. Write the warm-up payoff matrix and explain what its entry in row 2, column 2 means.
Multiply by 1/2: [3 -2; -1 1.5]. The entry in row 2, column 2 means that when the row player picks row 2 and the column player picks column 2, the row player wins 1.5 points in the warm-up round, half of the 3 points in a regular round.
Question 18 of 20 · Short Answer
Find k if k[8 -4; 12 20] = [6 -3; 9 15].
k = 3/4. Divide matching entries: 6/8 = 3/4, -3/(-4) = 3/4, 9/12 = 3/4 and 15/20 = 3/4. All four agree, so 0.75 times the matrix gives the result.
Question 19 of 20 · Short Answer
A matrix A satisfies 2A = [6 -4; 10 8]. Find A, then find -3A.
Multiply both sides by 1/2: A = [3 -2; 5 4]. Then -3A = [-9 6; -15 -12]. Students can check the second answer another way: -3A is -3/2 times 2A, and -3/2 times [6 -4; 10 8] gives the same matrix.
Question 20 of 20 · Short Answer
A gym's monthly fees in dollars have rows adult and student and columns basic and premium: [30 45; 20 35]. During a promotion, all fees are 10% off. Write the promotion fees as a scalar multiple and compute them. Why is the answer not the original matrix minus 0.10?
A 10% discount leaves 90% of each fee, so the promotion matrix is 0.9 times the fee matrix = [27 40.5; 18 31.5], in dollars. Subtracting 0.10 would take 10 cents off each fee, not 10%, and adding or subtracting a single number from a matrix is not a defined matrix operation.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.VM.C.7 mean?
HSN.VM.C.7 means students can multiply a matrix by a scalar to make a new matrix. A scalar is a single real number, and multiplying a matrix by it multiplies every entry by that number. The standard gives the example of doubling all of the payoffs in a game. The code reads: High School (HS), Number and Quantity (N), Vector and Matrix Quantities (VM), cluster C, standard 7.
Is HSN.VM.C.7 taught in Algebra 2 or Precalculus?
It is usually taught in Precalculus. HSN.VM.C.7 is a (+) standard, which Common Core describes as additional mathematics for students who take advanced courses, so it is not required in a standard Algebra I or Algebra II course. Some honors Algebra II courses include a matrices unit that covers it.
How do you multiply a matrix by a scalar?
Multiply each entry of the matrix by the scalar and keep every entry in its place. For example, 5[2 0; -1 3] = [10 0; -5 15]. Working row by row helps students avoid skipping an entry.
What does "all of the payoffs in a game are doubled" mean?
It means the new payoff matrix is 2 times the old one. A payoff matrix lists what a player wins or loses for each pair of choices. If every win and every loss is worth twice as much, each entry doubles, and that is exactly the scalar multiple 2P. Negative payoffs stay negative, so a loss becomes a bigger loss.
Does multiplying by a scalar change the size of a matrix?
No. A scalar multiple has the same number of rows and columns as the original matrix. Only the entries change. This is one way students can check an answer quickly: if the dimensions changed, something went wrong.
What is the difference between scalar multiplication and matrix multiplication?
Scalar multiplication multiplies a matrix by one number, so every entry is multiplied by the same value. Matrix multiplication (HSN.VM.C.8) multiplies two matrices, combining rows of the first with columns of the second, and it has rules about which dimensions fit. Scalar multiplication works for a matrix of any size.
How do you find the scalar k when you know A and kA?
Divide an entry of kA by the matching nonzero entry of A, then check that the same k works for every other entry. For example, if k[2 5] = [7 17.5], then k = 7/2 = 3.5, and 3.5(5) = 17.5 confirms it. If different entries give different ratios, the second matrix is not a scalar multiple of the first.
How do you use a scalar for a percent increase or decrease?
Multiply by 1 plus the percent for an increase and by 1 minus the percent for a decrease, written as decimals. An 8% increase uses the scalar 1.08, and a 30% discount uses 0.7. Multiplying by 0.08 gives only the amount of the increase, a common error.
What are common mistakes with scalar multiplication?
Common mistakes include:
Multiplying only the first row or the first entry
Adding the scalar to each entry instead of multiplying
Sign errors with a negative scalar, especially on entries that are already negative
Dividing the wrong way when finding an unknown scalar
How does scalar multiplication of matrices connect to other standards?
It is the matrix version of multiplying a vector by a scalar (HSN.VM.B.5), where each component is multiplied by the same number. It builds on using matrices to organize data (HSN.VM.C.6) and leads to adding, subtracting and multiplying matrices (HSN.VM.C.8), where expressions like 2A + B combine scalar multiples. When the columns of a matrix are points, a positive scalar multiple is a dilation centered at the origin.
07
Related Standards
5 standards
These standards connect to HSN.VM.C.7: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.NS.A.2Prerequisite
Multiply and divide rational numbers, including negative numbers
