16 Multiplying Matrices Warm Up Lesson Presentation Lesson Quiz

Warm Up State the dimensions of each matrix. 1. [3 1 4 6] 2. Calculate. 3. 3(–4) + (–2)(5) + 4(7) 4. (–3)3 + 2(5) + (–1)(12) 1  4 3  2 6 –11

Objectives Understand the properties of matrices with respect to multiplication. Multiply two matrices.

Vocabulary matrix product square matrix

In the last lesson, you multiplied matrices by a number called a scalar. You can also multiply matrices together. The product of two or more matrices is the matrix product. The following rules apply when multiplying matrices. Matrices A and B can be multiplied only if the number of columns in A equals the number of rows in B. • The product of an m  n and an n  p matrix is an m  p matrix.

Determine whether the product of. is defined Determine whether the product of is defined. If so, state the dimensions of the product. A • B = AB 3  4 4  2 3  2 The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 3  2. Example 3-1a

Determine whether the product of. is defined Determine whether the product of is defined. If so, state the dimensions of the product. A • B 3  2 4  3 The inner dimensions are not equal, so the matrix product is not defined. Example 3-1b

Answer: The matrix product is defined. The dimensions are 3  3. Determine whether each matrix product is defined. If so, state the dimensions of the product. a. b. Answer: The matrix product is defined. The dimensions are 3  3. Answer: The matrix product is not defined. Example 3-1c

The CAR key: Columns (of A) As Rows (of B) or matrix product AB won’t even start Helpful Hint

Find RS if Step 1 Multiply the numbers in the first row of R by the numbers in the first column of S, add the products, and put the result in the first row, first column of RS. Example 3-2a

Step 2 Multiply the numbers in the first row of R by the numbers in the second column of S, add the products, and put the result in the first row, second column of RS. Example 3-2b

Step 3 Multiply the numbers in the second row of R by the numbers in the first column of S, add the products, and put the result in the second row, first column of RS. Example 3-2c

Step 4 Multiply the numbers in the second row of R by the numbers in the second column of S, add the products, and put the result in the second row, second column of RS. Example 3-2d

Step 5 Simplify the product matrix. Answer: So, Example 3-2e

Find WX Multiply row 1 of W and column 1 of X as shown. Place the result in wx11. 3(4) + –2(5)

Multiply row 1 of W and column 2 of X as shown Multiply row 1 of W and column 2 of X as shown. Place the result in wx12. 3(7) + –2(1)

Multiply row 1 of W and column 3 of X as shown Multiply row 1 of W and column 3 of X as shown. Place the result in wx13. 3(–2) + –2(–1)

Multiply row 2 of W and column 1 of X as shown Multiply row 2 of W and column 1 of X as shown. Place the result in wx21. 1(4) + 0(5)

Multiply row 2 of W and column 2 of X as shown Multiply row 2 of W and column 2 of X as shown. Place the result in wx22. 1(7) + 0(1)

Multiply row 2 of W and column 3 of X as shown Multiply row 2 of W and column 3 of X as shown. Place the result in wx23. 1(–2) + 0(–1)

Multiply row 3 of W and column 1 of X as shown Multiply row 3 of W and column 1 of X as shown. Place the result in wx31. 2(4) + –1(5)

Multiply row 3 of W and column 2 of X as shown Multiply row 3 of W and column 2 of X as shown. Place the result in wx32. 2(7) + –1(1)

Multiply row 3 of W and column 3 of X as shown Multiply row 3 of W and column 3 of X as shown. Place the result in wx33. 2(–2) + –1(–1)

Find RS if Answer: Example 3-2f

Chess Three teams competed in the final round of the Chess Club’s championships. For each win, a team was awarded 3 points and for each draw a team received 1 point. Find the total number of points for each team. Which team won the tournament? Team Wins Draws Blue 5 4 Red 6 3 Green Explore The final scores can be found by multiplying the wins and draws by the points for each. Example 3-3a

Plan Write the results from the championship and the points in matrix form. Set up the matrices so that the number of rows in the points matrix equals the number of columns in the results matrix. Results Points Example 3-3b

Solve Multiply the matrices. Write an equation. Multiply columns by rows. Example 3-3c

The labels for the product matrix are shown below. Simplify. The labels for the product matrix are shown below. Blue Red Green Total Points Example 3-3d

Answer: The blue team had 19 points, the red team had 21 points, and the green team had 17 points. The red team won the championship. Examine: R is a 3  2 matrix and P is a 2  1 matrix. Their product should be a 3  1 matrix. Example 3-3e

Basketball In Thursday night’s basketball game, three of the players made the points listed below in the chart. They scored 1 point for the free-throws, 2 points for the 2-point shots, and 3 points for the 3-points shots. How many points did each player score and who scored the most points? Player Free-throws 2-point 3-point Warton 2 3 Bryant 5 1 Chris 4 Answer: Warton scored 14 points, Bryant scored 7 points, and Chris scored 25 points. Chris scored the most points in the game. Example 3-3f

Multiply columns by rows. Find KL if Substitution Multiply columns by rows. Example 3-4a

Answer: Simplify. Example 3-4b

Multiply columns by rows. Find LK if Substitution Multiply columns by rows. Example 3-4c

Answer: Simplify. Example 3-4d

Find each product if a. AB b. BA Answer: Answer: Example 3-4e

Add corresponding elements. Find A(B + C) if Substitution Add corresponding elements. Example 3-5f

Multiply columns by rows. Answer: or Example 3-5g

Multiply columns by rows. Find AB + AC if Substitution Multiply columns by rows. Example 3-5h

Add corresponding elements. Answer: Simplify. Add corresponding elements. Answer: Example 3-5i

Find each product if a. A(B + C) b. AB + AC Answer: Answer:

Businesses can use matrix multiplication to find total revenues, costs, and profits.

Example: Inventory Application Two stores held sales on their videos and DVDs, with prices as shown. Use the sales data to determine how much money each store brought in from the sale on Saturday. Use a product matrix to find the sales of each store for each day.

On Saturday, Video World made $851.05 and Star Movies made $832.50. Example Continued Fri Sat Sun Video World Star Movies On Saturday, Video World made $851.05 and Star Movies made $832.50.

Skateboard Kit Inventory Check It Out! Example 3 Change Store 2’s inventory to 6 complete and 9 super complete. Update the product matrix, and find the profit for Store 2. Skateboard Kit Inventory Complete Super Complete Store 1 14 10 Store 2 6 9

Check It Out! Example 3 Use a product matrix to find the revenue, cost, and profit for each store. Revenue Cost Profit Store 1 Store 2 The profit for Store 2 was $819.

Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A2 4. BD 5. C3

Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A2 4. BD 5. C3 not possible not possible