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Math 166-copyright Joe Kahlig, 09C
Section 5.2: Matrix Multiplication
Multiplication of Matrices: Two matrices A and B can be multiplied such that AB = C provided
that the dimension of A is m × n and the dimension of B is n × r. The resultant matrix C will have
the dimension of m × r.
Example: Give the dimension of the following products(if the multiplication).
A
B
C
D
3×4 3×7 4×3 4×4
AB
AC
CB
CT D
ACB
Example: Find matrices A and B such that AB is possible and BA is not possible.
Example: Compute
"
1
4
2
5
3
6
#

7

 −1
0

−2

1 
−3
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Math 166-copyright Joe Kahlig, 09C
Example: Compute


7

 −1
0
−2 "
 1
1 
4
−3
Example: Compute
2
5
J2
3
6
#
where J =
"
x
2
1
0
#
Definition: An identity matrix, denoted I, is a square matrix (n × n) that has I(1,1) = I(2,2) =
... = I(n,n) = 1 and all of the other entries equal to zero. The identity matrix has the property that
IA = A and AI = A. (pick the size of the identity matrix so that the multiplication is possible.
Example: Compute
"
1
4
2
5
3
6
#

1

 0
0
0
1
0

0

0 
1
Example: Express the system of equations as a matrix equation.
3x + 2y = 7
x + 4y = 10
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Math 166-copyright Joe Kahlig, 09C
Example: Matrix A shown the number of servings of each food that each person ate each day. Matrix
B shows the number of units of fat, carbohydrates and protein in a serving of each food.
A=
Sandy
David
Bob





Food A
3
0
2
Food B
1
5
2

Food C

2



6
0

Food A 


Food B 
Food C
B=
fat
25
31
30
carbs
8
24
12
protein
12
19
22
Explain the meanings of the entries of

89
189
134
98

206 
132

72
192
64
99

227 
62
99

BA =  131
134
166

AB =  335
112


Example: The figure shows the connections(and directions of the connections) between different points.
The matrix A shows the number of paths from the row label to the column label.
B
A
A
A= B
C
D
D







A
0
1
1
0
B
1
0
0
0
C
0
0
0
1
D
1
1
0
0







C
A2
A
= B
C
D







A
1
0
0
1
B
0
1
1
0
C
1
1
0
0
D
1
1
1
0







A3
A
= B
C
D







A
1
2
1
0
B
1
0
0
1
C
1
1
1
0
D
1
1
1
1











