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HSN.VM.C.8Common CoreMathNumber and QuantityGrades 9-12

HSN.VM.C.8: Adding, Subtracting and Multiplying Matrices

In plain English: HSN.VM.C.8 is an advanced (+) Common Core number and quantity standard, usually taught in Precalculus, that asks students to add, subtract and multiply matrices of appropriate dimensions. Sums and differences are found entry by entry and need matrices of the same size; a product AB uses rows of A times columns of B and is defined only when A has as many columns as B has rows.

(+) Add, subtract, and multiply matrices of appropriate dimensions.

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.8 or N-VM.8 · Official standard
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01

Lesson Plan

70 min

Overview

Students learn the three matrix operations named in the standard and the dimension rule that goes with each. To add or subtract two matrices, they must have the same dimensions, and students combine matching entries. To multiply A by B, the number of columns of A must equal the number of rows of B. Then entry (i, j) of AB is found by multiplying the entries of row i of A by the matching entries of column j of B and adding the products, and an m × n matrix times an n × p matrix gives an m × p matrix.

Students decide first whether an operation is defined, then compute, then interpret the result in a context such as store inventory, ticket sales or the revenue from a set of orders. Throughout this page, matrices are written row by row, with rows separated by semicolons: [2 -1; 3 4] has first row 2, -1 and second row 3, 4, and [5; 14] is a column with 5 above 14.

Learning Objectives

By the end of this lesson, students will be able to:

  • Decide whether a sum, difference or product of two matrices is defined from their dimensions, and give the dimensions of the result
  • Add and subtract matrices of the same dimensions entry by entry
  • Multiply matrices by the row-by-column rule, including non-square matrices and a row times a column
  • Interpret sums, differences and products of data matrices in context, including what each entry of a product means

Prior Knowledge Required

Students should already be comfortable with:

  • Operations with positive and negative rational numbers 7.NS.A.2
  • Using matrices to organize data HSN.VM.C.6
  • Multiplying a matrix by a scalar HSN.VM.C.7
  • Matrix dimensions (rows × columns) and naming entry (i, j)

Lesson Procedure

70-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Post two ticket tables from a school play and ask two questions:

    Warm-Up Prompt

    "On Friday the play sold 40 adult and 25 student tickets online and 30 adult and 15 student tickets at the door. On Saturday it sold 35 adult and 20 student tickets online and 25 adult and 30 student at the door. (1) Make one table of the totals for both nights. (2) At the Friday matinee, 45 adults paid $8 each and 30 students paid $5 each. What was the revenue? Which numbers did you multiply together, and which did you add?"

    For (1), students add matching cells: with rows online and door and columns adult and student, [40 25; 30 15] + [35 20; 25 30] = [75 45; 55 45]. For (2), 45(8) + 30(5) = 360 + 150 = 510 dollars. Point out that (2) pairs each count with its price, multiplies the pairs and adds the products. That "pair, multiply, add" pattern is how every entry of a matrix product is computed.

  2. Direct Instruction25 minutes

    Part 1: Addition and subtraction. Two matrices can be added or subtracted only when they have the same dimensions. The sum A + B has entry aij + bij in position (i, j), and A - B has entry aij - bij. Stress subtraction signs: 3 - (-2) = 5.

    Part 2: Multiplication. Use Diagram 1 for the dimension check and Diagram 2 for the row-by-column rule, then give the procedure:

    1. Write the dimensions side by side: A is m × n and B is n2 × p. The product AB is defined only if n = n2 (the inner numbers match).
    2. Read the size of the answer: the outer numbers give AB its dimensions, m × p.
    3. Compute one entry at a time: for entry (i, j), take row i of A and column j of B, multiply first with first, second with second and so on, and add the products.
    4. Fill the answer row by row: finish row 1 of AB using row 1 of A with each column of B, then move to row 2.
    5. Interpret: in a context, the units of AB come from the row labels of A and the column labels of B.

    • Adding data matrices

      A school store sold hoodies (row 1) and T-shirts (row 2) in sizes small and medium (columns). September sales were [12 8; 5 10] and October sales were [9 11; 7 4]. Find the two-month totals.

      Equation: [12 8; 5 10] + [9 11; 7 4] = [21 19; 12 14]

    • Subtracting with negative entries

      Compute [5 -2 0; 3 7 -4] - [1 4 -6; -2 7 3].

      Equation: [4 -6 6; 5 0 -7]

    • Deciding what is defined

      Let A = [1 2; 3 4] (2 × 2) and B = [1 2 3; 4 5 6] (2 × 3). Which of A + B, AB and BA are defined?

      Equation: A + B is not defined (different sizes); AB is defined and 2 × 3; BA is not defined (3 columns, 2 rows)

    • Product of 2 × 2 matrices

      Compute [2 -1; 3 4][5 0; -2 1] using rows of the first matrix times columns of the second.

      Equation: [12 -1; 7 4]

    • Product in context

      A cafe sold coffees, bagels and muffins on Monday and Tuesday: Q = [40 25 15; 35 30 20]. Matrix V lists each item's price and cost in dollars: V = [3 1; 2.5 0.75; 3.5 1.25]. Find QV.

      Equation: QV = [235 77.5; 250 82.5]: row 1 is Monday's revenue and cost, row 2 is Tuesday's

    For the last example, ask why V has to list items in the same order as the columns of Q: each product pairs the number sold of an item with that same item's price. Then ask what VQ would mean. V is 3 × 2 and Q is 2 × 3, so VQ is defined and 3 × 3, but its entries multiply prices by numbers of different items and have no useful meaning. A product can be defined and still not answer the question you are asking.

  3. Guided Practice15 minutes

    Pairs work four items, and one partner explains the dimension check aloud before computing: (a) [3 -1; 0 6] + [-4 2; 5 -3] (answer: [-1 1; 5 3]); (b) [7 0 -2] - [3 -5 4] (answer: [4 5 -6]); (c) [1 3; -2 0][4 -1; 2 5] (answer: [10 14; -8 2]); (d) A is 3 × 2 and B is 2 × 4: is AB defined? BA? A + B? (Answer: AB is 3 × 4; BA and A + B are not defined.) Listen for these errors: multiplying matching entries instead of rows by columns, adding the entry products in the wrong row, sign errors in subtraction, and reading the result size from the inner numbers.

  4. Independent Practice15 minutes

    Students work alone on six items: (1) [2.5 -1; 4 0] + [1.5 3; -6 2] (answer: [4 2; -2 2]); (2) [0 8; -3 5; 1 1] - [2 8; 4 -1; -1 6] (answer: [-2 0; -7 6; 2 -5]); (3) [1 0 2][3; -1; 4] (answer: [11], a 1 × 1 matrix); (4) [2 1; -1 3][1 2 0; 4 -1 3] (answer: [6 3 3; 11 -5 9]); (5) with A 2 × 3 and B 3 × 3, which of AB and BA are defined? (answer: AB is 2 × 3; BA is not defined); (6) a bakery sold 10 loaves and 6 pies on Saturday and 4 loaves and 12 pies on Sunday, with loaves at $5 and pies at $14: compute [10 6; 4 12][5; 14] and say what each entry means (answer: [134; 188], the revenue in dollars on each day).

  5. Closure5 minutes

    Exit ticket: (1) Compute [4 -2; 1 3] - [6 -2; -5 0]. (Answer: [-2 0; 6 3].) (2) What are the dimensions of the product of a 4 × 2 matrix and a 2 × 5 matrix? (4 × 5.) (3) Without computing the whole product, find the entry in row 2, column 1 of [1 2; 3 -1][0 5; 4 2]. (3(0) + (-1)(4) = -4.)

Differentiation Strategies

For Struggling Students

  • Have students highlight a row of the first matrix in one color and a column of the second in another before computing each product entry
  • Give a dimension check template: write "m × n times n × p", circle the inner numbers, and box the outer numbers before any arithmetic
  • Start products with a row times a column, such as [2 3][4; 1] = [11], before moving to 2 × 2 products

For Advanced Students

  • Ask students to find matrices A and B for which both AB and BA are defined but have different dimensions
  • Ask students to write a data context in which a product of a 2 × 3 and a 3 × 2 matrix has a clear meaning for every entry, and trade contexts with a partner
  • Ask students to explain why a matrix A can be multiplied by itself only if A is square

Assessment Guidance

What to Look For

Listen for the dimension check before any computation: students should name the inner and outer numbers and say whether the operation is defined. For sums and differences, check that each entry combines the matching entries, with careful signs when subtracting negatives. For products, look for row-by-column work and a correct result size; a student who multiplies matching entries has confused the product with an entry-by-entry operation. In context problems, students should say what one entry of a product means, including units, and why the order of the factors matters for that meaning.

02

Classroom Activities

3 Activities

1

Dimension Detective

20 minPairs

Pairs receive 8 cards, each showing a matrix name and its dimensions. They find every sum and product that is defined, then compute a few with real entries. The activity builds the habit of checking dimensions first.

The 8 Cards

  • A (2 × 3), B (3 × 2), C (2 × 2), D (3 × 1)
  • E (1 × 3), F (2 × 3), G (3 × 3), H (1 × 2)

Procedure

  • List every pair of different cards whose sum is defined (key: only A + F)
  • List every product XY that is defined, where X and Y may be the same card, and give its dimensions. Key: there are 23, for example AB is 2 × 2, BA is 3 × 3, DE is 3 × 3, ED is 1 × 1 and HC is 1 × 2
  • Now use A = [1 0 2; -1 3 1], B = [2 1; 0 -1; 1 4] and F = [3 -2 0; 2 2 -5]. Compute AB, A + F and A - F (key: AB = [4 9; -1 0], A + F = [4 -2 2; 1 5 -4], A - F = [-2 2 2; -3 1 6])

Discussion Questions

  • Card G can be multiplied by itself, but card A cannot. Which cards can be squared, and what do they have in common?
  • AB and BA are both defined. Why do they have different dimensions?
  • Why are there many more defined products than defined sums?
2

School Store Inventory Tracker

15 minGroups of 3

Groups track a week of hoodie inventory with matrix addition and subtraction. Each group member is responsible for one matrix and explains what its entries mean.

The Data

  • Rows: navy, gray. Columns: sizes S, M, L, XL
  • Start of week S = [6 10 9 4; 5 8 7 3]
  • Shipment received D = [12 12 12 6; 6 6 6 6]
  • Hoodies sold L = [9 14 11 5; 7 10 9 4]

Procedure

  • Compute S + D, the stock after the shipment (key: [18 22 21 10; 11 14 13 9])
  • Compute S + D - L, the stock at the end of the week (key: [9 8 10 5; 4 4 4 5])
  • Circle every entry below 5 and write a reorder note for the store manager

Challenge Variation

The store also sells caps in one size. Could the cap counts be added to S as a matrix? Groups explain in terms of dimensions, then design a matrix layout that could hold hoodies and caps together.

3

Fundraiser Order Totals

20 minGroups of 3-4

Groups use a matrix product to find each homeroom's sales and profit in a school fundraiser, and then test the other order of multiplication.

The Data

  • Order matrix Q: rows are Homerooms 1, 2 and 3; columns are candles, cookie dough and wrapping paper. Q = [8 12 5; 10 6 9; 4 15 7]
  • Value matrix V: rows are the same three items; columns are sale price and school profit per item in dollars. V = [15 6; 12 5; 10 4]

Procedure

  • Check the dimensions of QV before computing (3 × 3 times 3 × 2 gives 3 × 2)
  • Each group member computes one row of QV and explains it to the group (key: QV = [314 128; 312 126; 310 127])
  • Decide which homeroom raised the most profit and which sold the most dollars of items
  • Try VQ and explain why it is not defined

Discussion Questions

  • What does the entry in row 2, column 1 of QV mean, with units?
  • Homeroom 2 sold more dollars of items than Homeroom 3 but earned less profit. Which items explain the difference?
  • If the school adds a fourth item, how must Q and V change so that QV is still defined?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: When Is a Matrix Product Defined?

Check the dimensions before you multiply A 2 × 3 B 3 × 2 = AB 2 × 2 inner: 3 = 3 outer numbers: size of AB Defined: each row of A has 3 entries and each column of B has 3 entries to pair with. A 2 × 3 C 2 × 3 inner: 3 ≠ 2 = not defined A row of A has 3 entries but a column of C has only 2. The sum A + C is defined: same size.
Write the dimensions side by side. A 2 × 3 matrix times a 3 × 2 matrix is defined because the inner numbers match, and the outer numbers give the size of the product, 2 × 2. A 2 × 3 matrix times another 2 × 3 matrix is not defined, although their sum is.

Diagram 2: The Row-by-Column Rule

Entry (1, 1) of AB: row 1 of A times column 1 of B 2 -1 3 4 5 0 -2 1 = 12 -1 7 4 A (2 × 2) B (2 × 2) AB (2 × 2) Row 1 of A: 2, -1 Column 1 of B: 5, -2 2(5) + (-1)(-2) = 10 + 2 = 12 Pair first with first, second with second. Entry (i, j) of AB uses row i of A and column j of B. Entry (2, 1): 3(5) + 4(-2) = 7. Entry (1, 2): 2(0) + (-1)(1) = -1. Entry (2, 2): 3(0) + 4(1) = 4.
Each entry of AB comes from one row of A and one column of B. Entry (1, 1) uses row 1 of A and column 1 of B: 2(5) + (-1)(-2) = 12. The other three entries follow the same pattern.

04

Homework Assignment

~30 min

HSN.VM.C.8 Homework: Matrix Addition, Subtraction and Multiplication

Directions: Before each computation, write the dimensions of the matrices and say whether the operation is defined. Show the products and sums for at least two entries of every matrix product. In Part 3, say what the entries of your answers mean.

Part 1: Adding and Subtracting (Problems 1-2)

  1. Compute: (a) [-3 5; 2 0.5] + [4 -5; -7 1.5] (b) [9 -4 1; 0 2 -8] - [3 -4 -2; 5 -1 0]
  2. Find the matrix X that makes X + [2 -1; 4 3] = [5 6; -2 3] true, and check your answer by adding. Then explain why [1 2 3] + [1; 2; 3] is not defined.

Part 2: Multiplying (Problems 3-4)

  1. Compute [3 -2; 1 4][2 5; -1 0]. Show the row-by-column work for every entry.
  2. Let A = [1 -1 2; 0 3 1] and B = [4 0; -2 1; 1 5]. Give the dimensions of AB and BA, then compute both products.

Part 3: Matrices in Context (Problems 5-6)

  1. A theater sold tickets on two weekends. Rows are matinee and evening; columns are adult, student and senior. Weekend 1: [45 60 20; 120 85 30]. Weekend 2: [52 48 25; 110 95 28]. (a) Find the total sales matrix. (b) Find Weekend 2 minus Weekend 1 and explain what a negative entry means.
  2. Two soccer teams order jerseys, shorts and socks: Q = [15 15 30; 12 12 24] (rows: Team 1, Team 2). Matrix C gives the price per item in dollars at two suppliers: C = [22 20; 14 15; 4 5] (rows: jerseys, shorts, socks; columns: Supplier X, Supplier Y). Compute QC, explain what each entry means, and decide which supplier each team should use.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Dimension CheckDimensions written and defined or not defined decided correctly every timeOne decision missing or wrongNo dimension checks
Addition and SubtractionAll entries correct, including signsOne or two entry errorsSeveral errors or wrong method
MultiplicationRow-by-column work shown; all entries and result size correctCorrect method with arithmetic errorsEntry-by-entry products or no work
InterpretationEntries explained with units; supplier choice justifiedExplanation incompleteMissing or incorrect

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

  1. Question 1 of 20 · Multiple Choice

    Compute [2 -3; 1 4] + [-5 3; 6 -1].

  2. Question 2 of 20 · Multiple Choice

    Compute [6 0; -2 5] - [4 -3; -2 8].

  3. Question 3 of 20 · Multiple Choice

    Which sum is defined?

  4. Question 4 of 20 · Multiple Choice

    A is a 3 × 4 matrix and B is a 4 × 2 matrix. What are the dimensions of AB?

  5. Question 5 of 20 · Multiple Choice

    A and B are both 3 × 2 matrices. What can you say about AB?

  6. Question 6 of 20 · Multiple Choice

    Compute [1 2; 0 -1][3 1; 2 4].

  7. Question 7 of 20 · Multiple Choice

    What is the entry in row 1, column 2 of [2 0 -1; 3 1 4][1 5; 2 -2; 0 3]?

  8. Question 8 of 20 · Multiple Choice

    Compute [1 -2 3][2; 0; -1].

  9. Question 9 of 20 · Multiple Choice

    A food truck's revenue matrix for two locations (rows) and two weeks (columns) is R = [500 620; 480 700], in dollars. Its cost matrix is C = [320 410; 300 450]. Which matrix gives the profit?

  10. Question 10 of 20 · Multiple Choice

    Matrix Q (3 × 4) lists how many of 4 items each of 3 stores sold. Matrix P (4 × 1) lists the price of each item. What does QP give?

  11. Question 11 of 20 · Multiple Choice

    Find x and y if [x 2; 1 3] + [4 -1; y 0] = [7 1; -2 3].

  12. Question 12 of 20 · Multiple Choice

    A student computes [1 2; 3 4][2 0; 1 5] and writes [2 0; 3 20]. What is the correct product?

  13. Question 13 of 20 · Multiple Choice

    Let A = [2 1; 0 3]. What is A², the product AA?

  14. Question 14 of 20 · Multiple Choice

    Which product is defined?

  15. Question 15 of 20 · Short Answer

    Compute [0.5 -2; 3 1] - [-1.5 4; 3 -2].

  16. Question 16 of 20 · Short Answer

    Compute [3 0 -1; 2 1 1][1 2; -1 0; 4 1]. State the dimensions of the product first.

  17. Question 17 of 20 · Short Answer

    Explain why [2 0; -1 3] + [5 6] is not defined. Is the product [5 6][2 0; -1 3] defined? If so, compute it.

  18. Question 18 of 20 · Short Answer

    Two cafes record daily drink sales. Rows are coffee and tea; columns are small and large. Cafe 1: [34 21; 12 9]. Cafe 2: [28 30; 15 6]. Find the combined sales matrix and say what its entry in row 1, column 2 means.

  19. Question 19 of 20 · Short Answer

    At lunch, Ana buys 3 sandwiches and 2 drinks, and Ben buys 1 sandwich and 4 drinks. Sandwiches cost $6.50 and drinks cost $2.25. Write a matrix product that gives each person's total, and compute it.

  20. Question 20 of 20 · Short Answer

    A is 2 × 4, B is 4 × 3 and C is 3 × 2. For each of AB, BA, (AB)C and A + B, say whether it is defined and, if it is, give its dimensions.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does HSN.VM.C.8 mean?

HSN.VM.C.8 means students can add, subtract and multiply matrices whenever the dimensions allow it. The phrase "of appropriate dimensions" is the key: sums and differences need matrices of the same size, and a product AB needs A to have as many columns as B has rows. The code stands for High School, Number and Quantity, Vector and Matrix Quantities, cluster C, standard 8.

Is HSN.VM.C.8 Algebra 2 or Precalculus?

HSN.VM.C.8 is usually taught in Precalculus. It is a (+) standard, which Common Core describes as additional mathematics for students who take advanced courses. Some honors Algebra II courses teach matrix operations in a unit on systems of equations.

What does "appropriate dimensions" mean for matrices?

It means the sizes must fit the operation. For addition and subtraction, both matrices must be m × n. For multiplication, an m × n matrix can multiply an n × p matrix, and the product is m × p. For example, a 5 × 2 matrix times a 2 × 3 matrix gives a 5 × 3 matrix, but a 5 × 2 matrix times a 3 × 5 matrix is not defined.

Why don't you multiply matrices entry by entry?

Because matrix multiplication is defined to combine information, not to match positions. Each entry of AB pairs a row of A with a column of B, the way a count of items pairs with their prices to give a total. That definition is what makes products useful for totals in data problems and, later, for writing systems of equations and transformations.

How do you remember the row-by-column rule?

Read it as "row of the first, column of the second." Many teachers have students run a finger across a row of the first matrix while running another finger down a column of the second, multiplying as they go. For entry (2, 3), use row 2 and column 3.

Can you add a 2 × 3 matrix and a 3 × 2 matrix?

No. They have the same number of entries, but the entries are arranged differently, so entry (1, 3) of the first has no partner in the second. Only matrices with exactly the same dimensions can be added or subtracted.

Does the order of the matrices matter when you multiply?

Yes. AB can be defined when BA is not, and even when both are defined they can have different dimensions or different entries. Students study this in HSN.VM.C.9, which asks them to understand that matrix multiplication is not commutative even for square matrices.

Can students use a calculator for matrix operations?

Yes, once they can do small cases by hand. A graphing calculator or a free matrix app is helpful for 3 × 3 products and real data. Students should still check dimensions themselves and predict the size of the answer, because a calculator only reports a dimension error without explaining it.

What are common mistakes when multiplying matrices?

Common mistakes include:

  • Multiplying matching entries instead of rows times columns
  • Using a column of the first matrix instead of a row
  • Reading the result size from the inner numbers instead of the outer numbers
  • Computing a product that is not defined because the dimensions were never checked
How does HSN.VM.C.8 connect to other standards?

It builds on representing data in matrices (HSN.VM.C.6) and scalar multiplication (HSN.VM.C.7), and it parallels adding and subtracting vectors (HSN.VM.B.4). Matrix products lead directly to the properties of multiplication (HSN.VM.C.9), multiplying vectors by matrices (HSN.VM.C.11) and solving systems of equations with inverse matrices (HSA.REI.C.9).