Holt Algebra 2

4-2 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 2

4-2 Multiplying Matrices

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

Holt Algebra 2

4-2 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

4-2 Multiplying Matrices

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

P2 5 Q5 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 2

4-2 Multiplying Matrices

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

P2 5 Q5 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 2

4-2 Multiplying Matrices

Holt Algebra 2

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 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

4-2 Multiplying Matrices

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

Holt Algebra 2

4-2 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

4-2 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

4-2 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 2

4-2 Multiplying Matrices

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

Holt Algebra 2

4-2 Multiplying Matrices

Example 4A: Finding Powers of Matrices

Evaluate, if possible.

P3

Holt Algebra 2

4-2 Multiplying Matrices

Example 4A Continued

Holt Algebra 2

4-2 Multiplying Matrices

HW pg. 258

#’s 39, 41-45, 47, 51