Lesson 3: Multiplying Matrices. You can only multiply two matrices if

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1 Lesson 3: Multiplying Why Do We Multiply? When the Baltimore Ravens won the Super Bowl in 2013, Dan knew that his NFL Sports attire business would go crazy! He started to produce hats, shirts, sweatshirts, and scarves. He would sell the hats for $30, t-shirts for $35, sweatshirts for $45 and scarves for $30. Here s how he organized his information using matrices, where P is the amount produced and C is the cost per item. 100 hats 150 shirts P = 200 sweatshirts C = scarves We can use matrix multiplication to find the total revenue Dan would make if he sold everything that he produced. C * P = You can only multiply two matrices if When Can You Multiply? = =

2 Find AB if A = and B = How Do You Multiply Square? Step 1: Multiply the numbers in first row of A by the numbers in the first column of B, add the products and put the result in the first row, first column of AB = Step 2: Multiply the numbers in the first row of A by the numbers in the second column of B, add the products, and put the result in the first row, second column of AB = Step 3: Multiply the numbers in the second row of A by the numbers in the first column of B, add the products, and put the result in the second row, first column of AB = Step 4: Multiply the numbers in the second row of A by the numbers in the second column of B, add the products, and put the result in the second row, second column of AB = Step 5: Simplify the matrix. Guide to Multiplying =

3 How to Multiply with Different Dimensions = Pulling it All Together Given: A = B = 2 C Find: B(A+C). Then check the solution on your graphing calculator.

4 Does the Commutative Property of Multiplication Work with? Given: A = and B = Find: AB and BA.

5 Lesson 3: Multiplying Part 1: Determine whether each matrix is defined. (Can we multiply them together?) If they are defined, state the dimensions of the product. 1. R 2x3 S 3 x4 2. A 1x3 B 3 x 2 3. F 2x3 G 2x3 4. M 2x9 N 9 x1 5. Give an example of a matrix whose product is a 4 x2 matrix. 6. Give an example of a matrix whose product is a square matrix. 7. Give an example of a matrix whose product is a column matrix. Part 2: Find each product, if possible In the expression, AB, if A is a 3x4 matrix, what could be the dimensions of B if you want a defined product?

6 19. Jason has 3 Italian ice/custard stands. He sells Italian ice for $1.50, frozen custard for $2.25 and soft ice cream for $2.50. Each week, he tallies the number of sales for each product and organizes it in a table like the one below. Store Italian Ice Frozen Soft Ice Custard cream A B C Write a sales matrix for the number of each type of product sold for each store. Write a cost matrix for the price per item sold. Find the total income of the 3 items for each store expressed as a matrix. What is the total income from all three stores combined? 1. Give an example of two matrices whose product is a row matrix. 2. Is the matrix product defined? If so, explain why and state the dimensions. A 3x2 B 2x5 3. Find the product: Find the product:

7 Lesson 3: Multiplying Answer Key Part 1: Determine whether each matrix is defined. (Can we multiply them together?) If they are defined, state the dimensions of the product. 1. R 2x3 S 3 x4 2. A 1x3 B 3 x 2 R 2x3 S 3 x4 Since the inner dimensions are the same, the matrix product is defined. Its product will be a 2x4 matrix. A 1x3 B 3 x 2 Since the inner dimensions are the same, the matrix product is defined. Its product will be a 1x2 matrix. 3. F 2x3 G 2x3 4. M 2x9 N 9 x1 F 2x3 G 2x3 Since the inner dimensions are NOT the same, the matrix product is undefined. M 2x9 N 9 x1 Since the inner dimensions are the same, the matrix product is defined. Its product will be a 2x1 matrix. 5. Give an example of a matrix whose product is a 4 x2 matrix. Since the product matrix is a 4 x 2, that means that the outer dimensions must be 4 and 2 and the inner dimensions must be the same. There are many answers, here is one. A 4x3 B 3x2 Must be the same number (does not have to be 3) Must be 4 Must be 2 6. Give an example of a matrix whose product is a square matrix. A square matrix is a matrix that has the same number of columns and rows. Therefore, the following are square matrices: 2x2, 3x3, 4x4, 5x5. You can choose any of these. I will choose a 2x2 product. A 2x4 B 4 x2 Must be the same number (does not have to be 4) ` In order to be a square matrix, these two numbers must be the same (they don t have to be 2.)

8 7. Give an example of a matrix whose product is a column matrix. A column matrix has 1 column (the number of rows can vary). Examples of column matrices are: 2x1, 3x1, 4x1, 5x1, Therefore, the outer dimension of the second matrix must be 1, and the inner dimensions must be the same. A 2x4 B 4 x1 Must be a 1 Must be the same number Part 2: Find each product, if possible x2 2x2 I know this will be a 2x2 product 2x1 1x2 This will be a 2x = = = = x3 3x2 This will be a 2x2 3x1 1x2 This will be a 3x = =

9 x3 2x3 1x2 2x2 This will be a 1x2 This product is undefined because the inside dimensions are not the same = = x = 2x2 This will be a 2x2 product x x1 This product is undefined because the inside dimensions are not the same x x3 This will be a 3x3 product = = = 2x2 2x2 This will be a 2x2 product

10 18. In the expression, AB, if A is a 3x4 matrix, what could be the dimensions of B if you want a defined product.? If A is a 3x4 matrix, then B could be any matrix that has 4 rows. Therefore, B could be a 4x2 or it could be 4x3 or 4x4. (The inner dimensions (4) must match in order to have a defined product.) 19. Jason has 3 Italian ice/custard stands. He sells Italian ice for $1.50, frozen custard for $2.25 and soft ice cream for $2.50. Each week, he tallies the number of sales for each product and organizes it in a table like the one below. Store Italian Ice Frozen Soft Ice Custard cream A B C Write a sales matrix for the number of each type of product sold for each store = This is a 3x3 matrix because you have 3 stores and 3 products. Make sure you keep the numbers organized as they are in the table. Write a cost matrix for the price per item sold = This is a 3x1 matrix because the number of rows in the cost matrix must equal the number of columns in the sales matrix. So we end of up with SC. Find the total income of the 3 items for each store expressed as a matrix Cost Products = total income * 2.25 = We are multiplying a 3x3 by a 3x1, so this will give us a matrix product of 3x =

11 What is the total income from all three stores combined? In order to find the total income from all three stores, we need to add together the totals from the matrix above = The total income from all three stores combined is $ Give an example of two matrices whose product is a row matrix. (1 point) A row matrix has exactly one row. Therefore, a row matrix might be 1x2, or 1x3, or 1x4. An example is: A 1x4 * B 4x2. The product would be a 1x2 matrix, which is a row matrix. 2. Is the matrix product defined? If so, explain why and state the dimensions. (1 point) A 3x2 B 2x5 Yes, this matrix is defined because the number of columns of the first matrix is the same as the number of rows in the second matrix. (The inner dimensions are equal.) The product will be a 3x5 matrix. 3. Find the product: (2 points) = Find the product: (2 points) =