4 2 Multiplying Matrices Objectives Understand the properties

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

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

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

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. Holt Algebra 2

4 -2 Multiplying Matrices Find the product, if possible. WX Check the dimensions. W is 3 2 , X is 2 3. WX is defined and is 3 3. Multiply row 1 of W and column 1 of X as shown. Place the result in wx 11. 3(4) + – 2(5) Holt Algebra 2

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

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

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

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

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

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

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

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

4 -2 Multiplying Matrices 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. Holt Algebra 2

4 -2 Multiplying Matrices 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. Holt Algebra 2

4 -2 Multiplying Matrices 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. Holt Algebra 2

4 -2 Multiplying Matrices 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. Holt 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. Use a product matrix to find the sales of each store for each day. Holt Algebra 2

4 -2 Multiplying Matrices Fri Sat Sun Video World Star Movies On Saturday, Video World made $851. 05 and Star Movies made $832. 50. Holt 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. Because square matrices can be multiplied by themselves any number of times, you can find powers of square matrices. Holt Algebra 2

4 -2 Multiplying Matrices Evaluate, if possible. P 3 Holt Algebra 2

4 -2 Multiplying Matrices Check Use a calculator. Evaluate, if possible. Holt Algebra 2 Q 2

4 -2 Multiplying Matrices Check Use a calculator. Evaluate if possible. C 2 The matrices cannot be multiplied. Holt Algebra 2 A 3

4 -2 Multiplying Matrices Evaluate if possible. B 3 Holt Algebra 2 I 4