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

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nand an npmatrix is an mpmatrix.

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

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44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.

Check It Out! Example 1a Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 QP Q P 5325 The inner dimensions are not equal (3 ≠ 2), so the matrix product is not defined.

Check It Out! Example 1b Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 PQ P Q 2553 The inner dimensions are equal (5 = 5), so the matrix product will be a 2 x 3 matrix.

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 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 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.

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.

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.

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

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 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.

Example 4A: Finding Powers of Matrices Evaluate, if possible. P3

Example 4A Continued

HW pg. 258 #’s 39, 41-45, 47, 51

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