Holt Algebra 2

4-3 Multiplying Matrices

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.

Holt Algebra 2

4-3 Multiplying Matrices

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

Holt Algebra 2

4-3 Multiplying Matrices

The CAR key:Columns (of A)AsRows (of B)or matrix product ABwon’t even start

Helpful Hint

Holt Algebra 2

4-3 Multiplying Matrices

Tell whether the product is defined. If so, give its dimensions.

Example 1A: Identifying Matrix Products

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.

Holt Algebra 2

4-3 Multiplying Matrices

Tell whether the product is defined. If so, give its dimensions.

Example 1B: Identifying Matrix Products

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.

Holt Algebra 2

4-3 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

4-3 Multiplying Matrices

Holt Algebra 2

4-3 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 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 1 of W and column 1 of X as shown. Place the result in wx11.

3(4) + –2(5)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 1 of W and column 2 of X as shown. Place the result in wx12.

3(7) + –2(1)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 1 of W and column 3 of X as shown. Place the result in wx13.

3(–2) + –2(–1)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 2 of W and column 1 of X as shown. Place the result in wx21.

1(4) + 0(5)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 2 of W and column 2 of X as shown. Place the result in wx22.

1(7) + 0(1)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 2 of W and column 3 of X as shown. Place the result in wx23.

1(–2) + 0(–1)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 3 of W and column 1 of X as shown. Place the result in wx31.

2(4) + –1(5)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 3 of W and column 2 of X as shown. Place the result in wx32.

2(7) + –1(1)

Holt Algebra 2

4-3 Multiplying Matrices

Example 2A Continued

Multiply row 3 of W and column 3 of X as shown. Place the result in wx33.

2(–2) + –1(–1)

Holt Algebra 2

4-3 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 2

4-3 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.

Holt Algebra 2

4-3 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 2

4-3 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 2

4-3 Multiplying Matrices

240

531

14

28

240

531

14

28

22164

36168=

2x2 2x3 2x3

2x3 2x2

Product is not possible

Multiplication of matrices is not commutative!!!!!!

Holt Algebra 2

4-3 Multiplying Matrices

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

Holt Algebra 2

4-3 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 2

4-3 Multiplying Matrices

Example 3 Continued

On Saturday, Video World made $851.05 and Star Movies made $832.50.

Fri Sat SunVideo World

Star Movies

Holt Algebra 2

4-3 Multiplying Matrices

Assignment• Worksheet 4-3

• Pages 217-219 #’s 1-4, 6, 7, 10-12.

• Quiz on Sections 4-1 to 4-3 on Thursday!!!!