In Lesson 4-2, you multiplied matrices by a number called a scalar In Lesson 4-2, 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.

An m  n matrix A can be identified by using the notation Am  n.

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

Example 1A: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. A3  4 and B4  2; AB A B AB 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.

Example 1B: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. C1  4 and D3  4; CD C D 1  4 3  4 The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.

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.

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.

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

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

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

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

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

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

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

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

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

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.

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.

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.

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.

= 2x2 2x3 2x3 Product is not possible 2x3 2x2 Multiplication of matrices is not commutative!!!!!!

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

Example 3: 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.

Example 3 Continued Fri Sat Sun Video World Star Movies On Saturday, Video World made $851.05 and Star Movies made $832.50.

Assignment Worksheet 4-3 Pages 217-219 #’s 1-4, 6, 7, 10-12. Quiz on Sections 4-1 to 4-3 on Thursday!!!!