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3.6 Multiply Matrices
Goal  Multiply matrices.
Your Notes
Example 1
Describe matrix products
State whether the product AB is defined. If so, give the dimensions of AB.
a. A: 2  3, B: 4  3
b. A: 3  3, B: 3  2
a. Because the number of __columns__ in A (three) __does not equal__ the number
of __rows__ in B (four), the product AB __is not__ defined.
b. Because A is a 3  3 matrix and B is a 3  2 matrix, the product AB __is__ defined
and is a __3  2__ matrix.
MULTIPLYING MATRICES
Words To find the element in the ith row and jth column of the product matrix AB,
multiply each element in the __ith row of A__ by the corresponding element in
the __jth column of B__, then add the products.
A
Algebra
a b  


c d 
B
e

g
AB
f  ae  bg af  bh 


h   ce  dg cf  dh
Example 2
Find the product of two matrices
3  6 
1  2
Find AB if A  
and B  
 .

7  1 
5  4
Because A is a 2  2 matrix and B is a 2  2 matrix, the product AB __is__ defined and is
a __2  2__ matrix.
1  2 3  6
AB  


5  4 7  1
 1  3    2   7
 
5 3    4  7
  11  4 


 13  26
 1   6    2  
 5  6    4  
1 

1 
Your Notes
Checkpoint Given A and B, give the dimensions of AB. Then find AB.
 6
2


3,
1. A   1
 0  4
 1
2
B

 5  2
 4
8


3  2;  16  8
 20
8
9 
4  6
, B   
2. A  
3
7
2
24
2  1;  
69
Example 3
Use matrix operations
 4  3
3  5  2
 1
7
, B
If A  
, and C  

, evaluate each expression.

2


1
0
6
4
2
 1




a. (A + B)C b. AC + BC
Solution
  4  3  1
7  3  5  2

a. (A + B) C   

 

2  4  2  1
0
6
  1
 5 4 3  5  2   19  25 14 



0
6   15
25 10 
5 0  1
7  3  5  2
 4  3 3  5  2  1

b. AC + BC  





2  1
0
6    4  2  1
0
6
 1
 9  20  26  10  5 40



5
14  14 20  4
 1
 19  25 14

25 10
 15
Your Notes
PROPERTIES OF MATRIX MULTIPLICATION
Let A, B, and C be matrices and let k be a scalar.
Associative Property of Matrix Multiplication
Left Distributive Property
Right Distributive Property
Associative Property of Scalar Multiplication
A(BC) = __(AB)C__
A(B + C) = __AB + AC__
(A + B)C = __AC + BC__
k(AB) = __(kA)B_ = __A(kB)__
Example 4
Use matrices to calculate total cost
The school stores from the middle school and the high school each submit an inventory
list for the year. Each sweatshirt costs $15, each T-shirt costs $9, and each pennant costs
$5. Use matrix multiplication to find the total cost of the inventory for each school store.
Middle School: 61 sweatshirts, 63 T-shirts, and 74 pennants
High School: 58 sweatshirts, 71 T-shirts, and 92 pennants
Write the inventory and the cost in matrix form.
Cost
Dollars
Inventory
Sweatshirt T-shirts Pennant
63
74

Middle  61


High
71
92
 58

Sweatshirt
T-shirt
Pennant
 15 


 9 
 5 
Remember to set up
the matrices so that
the columns of the
inventory matrix
match the rows of
the cost matrix.
Find the total cost of inventory for each school store by multiplying the inventory matrix
by the cost matrix.
15
 61 63 74   61(15)  63(9)  74(5) 

  9  

58 71 92   58(15)  71(9)  92(5) 
5
 
1852 


1969 
Label the product matrix:
Middle School
High School
Total Cost Dollars
1852


1969
The total cost for the Middle School store is $ _1852_, and the total cost for the High
School store is $ _1969_.
Your Notes
Checkpoint Complete the following exercises.
1
 2
 3
  2  4


3. If A   3  2, B  
, and C   2, find A(BC).
1
 
 5
 0
4
  9


 20
 52
4. In Example 4,suppose that a sweatshirt costs $17,a T-shirt costs $11,and a pennant
costs $7.Calculate the total costs of inventory for each school store.
Middle School
High School
Total Cost Dollars
2248


2411
Homework
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