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
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
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
12/26/05 6:33:42 AM
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12/26/05 6:24:29 AM
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Holt Algebra 2
12/26/05 6:24:43 AM