11.2 Multiplying Matrices
Goal: Understand when it is possible to
multiply two or more matrices and find
the product of two or more matrices.
3
Expand:
log 7
5x
2 y4
Warm-up
 log 7 5 x3  log 7 2 y 4
 log 7 5  log 7 x3  (log 7 2  log 7 y 4 )
 log 7 5  3 log 7 x  (log 7 2  4 log 7 y )
 log 7 5  3 log 7 x  log 7 2  4 log 7 y
Condense:
3 log 4 2  log 4 x  2 log 4 7  3 log 4 y
 log 4 23  log 4 x  log 4 7 2  log 4 y 3
 log 4 8  log 4 x  log 4 49  log 4 y 3
 log 4 8 x  log 4 49 y 3
8x
 log 4
49 y 3
Matrix Multiplication
In order to multiply two matrices, the number of columns of the 1st matrix must equal
the number of rows of the 2nd matrix.
A  B  AB
m n n p
m p
Equal
Dimensions of AB
Given the dimensions of A and B, tell whether the product AB is defined. If so give
the dimensions of AB.
1. A: 2 X 3
B: 3 X 1
AB is defined. Its
dimensions are 2 X 1
2. A: 4 X 1
B: 4 X 1
AB is NOT defined.
Matrix Multiplication
Matrix Multiplication
To find the product of two matrices, multiply the elements of each
row of the first matrix by the elements of each column of the
second matrix, and then add the products.
Algebra:
a b   e
c d    g

 
f  ae  bg af  bh


h  ce  dg cf  dh
Example:
2 5 1 3  2(1)  5(0) 2(3)  5(2) 2  4
4 1  0  2   4(1)  1(0) 4(3)  1(2)   4 10 

 
 
 

Example
Multiply:
 3 4 1  2
 5 8  3 5 

 

3(1) + 4(3)


-5(1) + 8(3)

15
19


-5(-2) + 8(5)

3(-2) + 4(5)
14 

50
Example
Multiply:
 1 5   3  1
 4  2    2 6 

 

-1(3) + 5(-2) -1(-1) + 5(6)



4(3) + -2(-2) 4(-1) + -2(6)


 13
 16

31 

 16
Example
Multiply:
1(3) + 0(4)


-3(3) + 2(4)

6(3) + 1(4)


 1 0
  3 2   3 2 

 4  8
 6 1 


-3(2) + 2(-8)

6(2) + 1(-8) 

1(2) + 0(-8)
3
 1

 22
2 

 22

4 
Practice:
Find the product if it is defined:
1.
2 1 0
1 3 1

 
1
3
 
2.
6 8
1 39  8


Not defined
3.
 4
2 4 5
 
 16 20
  8 10 


4.
1 
2  3 56
2
 6
Properties of Matrix Multiplication
Properties of Matrix Multiplication
Let A, B, and C be matrices.
Associative Property of Matrix Multiplication
Left Distributive Property
Right Distributive Property
A( BC)  ( AB)C
A( B  C )  AB  AC
( A  B)C  AC  BC
Notice that there is no commutative property of matrix multiplication.
Example
 3  2
1
2  3
2


Let A  0
4 ,B
, and C  




1
0

4

2




 1 5 
Find AB  AC
AB  AC  A( B  C )
 3  2
0  4 
0

4 

 5 2 
 1 5 
3(0)  2(5) 3(4)  2(2) 
 0(0)  4(5) 0(4)  4(2) 


 1(0)  5(5)  1(4)  5(2)
 10  16
  20
8 


 25
14 
Assignment
Worksheet 11.2
2-14 evens
Click here for more
examples of multiplying
matrices.