Multiplying Matrices

Warm UpState the dimensions of each matrix.

1. [3 1 4 6]

2.

Calculate.

3. 3(–4) + (–2)(5) + 4(7)

4. (–3)3 + 2(5) + (–1)(12)

1 4

3 2

6

–11

Understand the properties of matrices with respect to multiplication.

Multiply two matrices.

Objectives

matrix productsquare matrixmain diagonalmultiplicative identity matrix

Vocabulary

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.

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

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

Helpful Hint

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.

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.

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

P2 5 Q5 3 R4 3 S4 5

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

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

P2 5 Q5 3 R4 3 S4 5

S R

4 5 4 3

Check It Out! Example 1b

SR

The inner dimensions are not equal (5 ≠ 4), so the matrix product is not defined.

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

P2 5 Q5 3 R4 3 S4 5

S Q

4 5 5 3

Check It Out! Example 1c

SQ

The inner dimensions are equal (5 = 5), so the matrix product is defined. The dimensions of the product are the outer numbers, 4 3.

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.

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

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.

Example 3 Continued

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

Fri Sat SunVideo World

Star Movies

Check It Out! Example 3

Change Store 2’s inventory to 6 complete and 9 super complete. Update the product matrix, and find the profit for Store 2.

Skateboard Kit Inventory

CompleteSuper

Complete

Store 1 14 10

Store 2 6 9

Check It Out! Example 3

Use a product matrix to find the revenue, cost, and profit for each store.

Revenue Cost ProfitStore 1Store 2

The profit for Store 2 was $819.

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.

Example 4A: Finding Powers of Matrices

Evaluate, if possible.

P3

Example 4A Continued

Example 4A Continued

Check Use a calculator.

Example 4B: Finding Powers of Matrices

Evaluate, if possible.

Q2

Check It Out! Example 4a

C2

Evaluate if possible.

The matrices cannot be multiplied.

Check It Out! Example 4b

A3

Evaluate if possible.

Check It Out! Example 4c

B3

Evaluate if possible.

Check It Out! Example 4d

I4

Evaluate if possible.

Lesson Quiz

Evaluate if possible.

1. AB

2. BA

3. A2

4. BD

5. C3

Lesson Quiz

Evaluate if possible.

1. AB

2. BA

3. A2

4. BD

5. C3

not possible

not possible