Reteach 4-2

Name LESSON

4-2

Date

Class

Reteach Multiplying Matrices

Use the dimensions to decide whether matrices can be multiplied. To multiply two matrices, the number of columns in A must equal the number of rows in B. Matrices:

A

Dimensions: m n

Remember, with matrices, AB is NOT the same as BA.

AB

B np

mp

Inner dimensions are equal: n n.

Outer dimensions give the dimensions of the product.

To determine which products are defined, check the dimensions. 1 2 3 5 1 1 2 C B 1 4 A 2 0 1 3 1 0 3 A: 2 3

B: 3 2

C: 2 2

AB: 2 3 and 3 2, so AB is defined and has dimensions 2 2. Inner dimensions are equal. AC: 2 3 and 2 2, so AC is not defined. Inner dimensions are NOT equal.

Use the following matrices for Exercises 13. Tell whether each product is defined. If so, give its dimensions. A

1

0

B

2 2

1. AB

3 1 2. BC

C 4 3 3. AC

A: 2 2

B:

2⫻1

A:

2⫻2

B: 2 1

C:

1⫻2

C:

1⫻2

Product defined?

Product defined?

Yes

yes

2⫻1

2⫻2

Copyright © by Holt, Rinehart and Winston. All rights reserved.

a207c04-2_rt.indd 14

14

Product defined?

no

Holt Algebra 2

12/26/05 6:33:42 AM Process Black

Name LESSON

Date

Class

Reteach

4-2 Multiplying Matrices (continued) To find a matrix product, first make sure the product is defined. 1 2 A

Find AB.

3 5

1

A is 2 3 and B is 3 2. The product is a 2 2 matrix.

B 1 4 0 3

2 0 1

Step 1: Multiply row 1 entries of A by column 1 entries of B. The sum is the first entry in the product. 1 2 3 5

1

1 4 2 0 1 0 3

3 1 5 1 1 0 ? ? ?

2 ? ? ?

Step 2: Multiply row 1 entries of A by column 2 entries of B. Add. 1 2 3 5

1

1 4 2 0 1 0 3

2 3 2 5 4 1 3 ?

?

2 29 ?

?

Step 3: Multiply row 2 entries of A by column 1 entries of B. Add. 1 2

2 29 2 29 1 4 2 ? 2 1 0 1 1 0 ? 2 0 1 0 3 3 5

1

Step 4: Multiply row 2 entries of A by column 2 entries of B. Add. 1 2

2 29 29 2 1 4 2 7 2 2 2 0 4 1 3 2 0 1 0 3 3 5

1

Find each product. 4.

3 4 1

3 3 1

4

3

3

4

1

3

12

9

4

1 2 1 1 0 4 5. 1 0 3 2 4 0 3 1 2 ⫺4 0 4 6. 5 2 0 3 1 2


a207c04-2_rt.indd 15

3 1 3

2

2

0

0

2

0

⫺1 ⫺2 11

6

2 16 3 6 15

Holt Algebra 2

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Holt Algebra 2

12/26/05 6:24:43 AM