Holt Algebra 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.

Holt Algebra Multiplying Matrices An m  n matrix A can be identified by using the notation A m  n.

Holt Algebra 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.

Holt Algebra Multiplying Matrices Tell whether the product is defined. If so, give its dimensions. P 2  5 Q 5  3 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

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

Holt Algebra Multiplying Matrices

Holt Algebra 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.

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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)

Holt Algebra 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.

Holt Algebra 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.

Holt Algebra 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.

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

Holt Algebra 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.

Holt Algebra Multiplying Matrices Example 3 Continued On Saturday, Video World made $ and Star Movies made $ Fri SatSun Video World Star Movies

Holt Algebra 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 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.

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

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

Holt Algebra Multiplying Matrices Example 4A Continued

Holt Algebra Multiplying Matrices HW pg. 258 #’s 39, 41-45, 47, 51