Page 1
Math 141-copyright Joe Kahlig, 10B
Section 2.5: 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).
B
C
D
A
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
Page 2
Math 141-copyright Joe Kahlig, 10B
Example: Compute


7

 −1
0
−2 "
 1
1 
4
−3
Example: Compute
2
5
3
6
J2
#
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

 0
0
"
1
4
2
5
3
6
#
"
1
4
2
5
3
6
#"
1
0

0
1
0
0
1
0

0 
1
#
Example: Express the system of equations as a matrix equation.
3x + 2y = 7
x + 4y = 10
Page 3
Math 141-copyright Joe Kahlig, 10B
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
B=

Food A 


Food B 
Food C
fat
25
31
30
carbs
8
24
12
Explain the meanings of the entries of
BA =

Food A 


Food B 
Food C
=
carbs
8
24
12
Sandy
David
Bob





protein
12
19
22

89
189
134
Food A
3
0
2
Food B
1
5
2

72
192
64
99

 131
134
=
AB =
fat
25
31
30
166

 335
112



Sandy 


∗


David 
Bob
Food A
3
0
2
Food B
1
5
2

Food C

2



6
0

98

206 
132

Food C

2
Food A

∗

6
Food B
0
Food C

99

227 
62





fat
25
31
30
carbs
8
24
12
protein
12
19
22





protein
12
19
22




