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Holt McDougal Algebra 2 4-2 Multiplying Matrices 4-2 Multiplying Matrices Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

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4-2 Multiplying Matrices 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

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Understand the properties of matrices with respect to multiplication. Multiply two matrices. Objectives

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Holt McDougal Algebra 2 4-2 Multiplying Matrices matrix product square matrix main diagonal multiplicative identity matrix Vocabulary

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Holt McDougal Algebra 2 4-2 Multiplying Matrices In Lesson 4-1, 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.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices An m n matrix A can be identified by using the notation A m n.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices The CAR key: Columns (of A) As Rows (of B) or matrix product AB wont even start Helpful Hint

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. Example 1A: Identifying Matrix Products A 3 4 and B 4 2 ; AB A BAB 3 4 4 2 = 3 2 matrix The inner dimensions are equal (4 = 4), so the matrix product is defined. The dimensions of the product are the outer numbers, 3 2.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. Example 1B: Identifying Matrix Products C 1 4 and D 3 4 ; CD C D 1 4 3 4 The inner dimensions are not equal (4 3), so the matrix product is not defined.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. P 2 5 Q 5 3 R 4 3 S 4 5 Q P 5 3 2 5 The inner dimensions are not equal (3 2), so the matrix product is not defined. Check It Out! Example 1a QP

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. P 2 5 Q 5 3 R 4 3 S 4 5 S R 4 5 4 3 Check It Out! Example 1b SR The inner dimensions are not equal (5 4), so the matrix product is not defined.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. P 2 5 Q 5 3 R 4 3 S 4 5 S Q 4 5 5 3 Check It Out! Example 1c SQ The inner dimensions are equal (5 = 5), so the matrix product is defined. The dimensions of the product are the outer numbers, 4 3.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Just as you look across the columns of A and down the rows of B to see if a product AB exists, you do the same to find the entries in a matrix product.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A: Finding the Matrix Product Find the product, if possible. WX Check the dimensions. W is 3 2, X is 2 3. WX is defined and is 3 3.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 1 of W and column 1 of X as shown. Place the result in wx 11. 3(4) + –2(5)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 1 of W and column 2 of X as shown. Place the result in wx 12. 3(7) + –2(1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 1 of W and column 3 of X as shown. Place the result in wx 13. 3(–2) + –2(–1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 2 of W and column 1 of X as shown. Place the result in wx 21. 1(4) + 0(5)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 2 of W and column 2 of X as shown. Place the result in wx 22. 1(7) + 0(1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 2 of W and column 3 of X as shown. Place the result in wx 23. 1(–2) + 0(–1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 3 of W and column 1 of X as shown. Place the result in wx 31. 2(4) + –1(5)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 3 of W and column 2 of X as shown. Place the result in wx 32. 2(7) + –1(1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2A Continued Multiply row 3 of W and column 3 of X as shown. Place the result in wx 33. 2(–2) + –1(–1)

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2B: Finding the Matrix Product Find each product, if possible. XW Check the dimensions. X is 2 3, and W is 3 2 so the product is defined and is 2 2.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 2C: Finding the Matrix Product Find each product, if possible. XY Check the dimensions. X is 2 3, and Y is 2 2. The product is not defined. The matrices cannot be multiplied in this order.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 2a Find the product, if possible. BC Check the dimensions. B is 3 2, and C is 2 2 so the product is defined and is 3 2.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 2b Find the product, if possible. CA Check the dimensions. C is 2 2, and A is 2 3 so the product is defined and is 2 3.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Businesses can use matrix multiplication to find total revenues, costs, and profits.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices 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. Example 3: Inventory Application Use a product matrix to find the sales of each store for each day.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 3 Continued On Saturday, Video World made $851.05 and Star Movies made $832.50. Fri SatSun Video World Star Movies

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 3 Change Store 2s 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 11410 Store 269

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Holt McDougal Algebra 2 4-2 Multiplying Matrices 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.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. The multiplicative identity matrix is any square matrix, named with the letter I, that has all of the entries along the main diagonal equal to 1 and all of the other entries equal to 0.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Because square matrices can be multiplied by themselves any number of times, you can find powers of square matrices.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 4A: Finding Powers of Matrices Evaluate, if possible. P 3

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 4A Continued

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 4A Continued Check Use a calculator.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Example 4B: Finding Powers of Matrices Evaluate, if possible. Q 2

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 4a C2C2 Evaluate if possible. The matrices cannot be multiplied.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 4b A3A3 Evaluate if possible.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 4c B3B3 Evaluate if possible.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Check It Out! Example 4d I4I4 Evaluate if possible.

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A 2 4. BD 5. C 3

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Holt McDougal Algebra 2 4-2 Multiplying Matrices Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A 2 4. BD 5. C 3 not possible

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