Matrices - Multiplication
• Assume that matrix A is of order m  n and matrix B is
of order p  q. To determine whether or not A can be
multiplied times B, write the matrices with their orders ...
A
mn
B
p  q.
• If the two inside numbers are the same, then matrix
multiplication is possible.
• When multiplication is possible, the resulting matrix
will have an order determined by the outside numbers.
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Matrices - Multiplication
• Example 1:
Find AB where A and B are given by ...
0 2  1
 1
 1  2 3


B   3  2 1 0
A

0  1
4
 2  1 0 3
23
34
• Since the inside numbers are the same, the multiplication
is possible.
• The resulting matrix will be 2  4.
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Slide 2
Matrices - Multiplication
• The process of multiplying is as follows:
 1  2 3
AB  

0  1
4
• To get the first entry of the
product matrix, note that it is
the row 1 column 1 entry.
• Multiply row 1 of matrix A
times column 1 of matrix B.
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0 2  1
 1
 3  2 1 0


 2  1 0 3
_

_
_
_
_
_
_

_
24
Slide 3
Matrices - Multiplication
 1  2 3
AB  

0  1
4
0 2  1
 1
 3  2 1 0


 2  1 0 3
• Multiply pairs of numbers by moving across the row
and down the column, and add the products.
(1)(1) + (-2)(3) + (3)(-2)
= 1
- 6
6
= -11
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 11 _
 _ _

_
_
_

_
Slide 4
Matrices - Multiplication
 1  2 3
AB  

0  1
4
• The next entry of the
product matrix is in row 1
and column 2.
0 2  1
 1
 3  2 1 0


 2  1 0 3
 11 _1 _

 _ _ _
_

_
• Multiplying as before with row 1 of matrix A
and column 2 of matrix B ...
(1)(0) + (-2)(-2) + (3)(-1) = 1
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Slide 5
Matrices - Multiplication
 1  2 3
AB  

0  1
4
The row 1 column 3 entry is ...
The row 1 column 4 entry is ...
The row 2 column 1 entry is ...
0 2  1
 1
 3  2 1 0


 2  1 0 3
_ 8_ 
 11 1 0


 6_ _1 _8 _7
The row 2 column 2 entry is ...
The row 2 column 3 entry is ...
The row 2 column 4 entry is ...
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Slide 6
Matrices - Multiplication
• Thus, the product of the matrices is ...
 1  2 3
AB  

0  1
4
0 2  1
 1
 3  2 1 0


 2  1 0 3
 11 1 0 8 


 6 1 8  7
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Slide 7
Matrices - Multiplication
• Example 2:
Find CD where C and D are given by ...
2
 1


C 3
9
 4  3
• The answer is ...
1
2  4
D

8  9
3
20  19
 4


CD   33
60  78
 1  40
31
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Slide 8
Matrices - Multiplication
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