Math 225 · 26 Sept 2013 · Discussion Questions
1 0 1
Exercise 1. (a) Calculate:
1 0 0
1 1 1 0
(b) Calculate:
=
0 0 1 0
1
=
0
(c) The two examples above illustrate that matrix multiplication differs from scalar multiplication in what
important way?
3 0 1 0
1 0 3 0
(d) Calculate
and
. Are they the same?
0 4 0 2
0 2 0 4
1 1 −2 −2
Exercise 2. (a) Calculate:
=
2 2
2
2
(b) The example above illustrates that matrix
important way?



a b c
0 0 0
(c) Calculate: d e f  0 0 0
g h i
0 0 0


 
a b c
0 0 0
0
(d) Is it true that d e f  0 0 0 = 0
g h i
0 0 0
0
Exercise

0
(a) 1
1
1
(b)
2
1
(c)
2
multiplication differs from scalar multiplication in what

0
a
0 d
0
g
0
0
0
b
e
h

c
f ?
i
3. Determine whether the following sets of matrices are linearly independent.
 

1 0
0 1 1
0 1 , 1 0 1
0 0
0 0 0
2
1 1
2 2
2 1
,
,
,
1
2 2
1 2
1 2
2
1 1
2 2
2 3
,
,
,
1
2 2
1 2
1 1
Exercise 4. Evaluate each of the following. Hint: by cleverly using properties of matrix algebra (for
example, aB + aC = a(B + C)), you can save yourself some work, but do make it painfully clear how you’re
doing your calculations.





 


1 2 3 1 0 0
1 2 3
8 9 8
1 2 3
−8 −9 −8
(a) 4 5 6 0 1 0
(e) 4 5 6 9 8 9+4 5 6 −9 −8 −9
7 8 9 0 0 1
1 2 3
8 9 8
1 2 3
−8 −9 −8
3 4 1 0
0 0
(b)
5 6 1 0
1 1




7
4
7
−7 −4 −7
2
7
(c) 3.75  5 −2 −7 + 3.75 −5



T T T
−2 −2 8
2 102 −8
 1 1 0 0  



(f) 
 0 1 1 1  
1 2
1 2
0 1 1 0
(d) 2.7
− 0.7
3 4
3 4
1

0
0

(g) 
0
9
0
0
0
9
0
0
0
2
1
0
1
0
2
5
0
0

 7
3 
8

9
 0

1 
0
0 0
0
7
5
8
0
0
0
1
0
0
9
8
0
0
1
0
5
8
0

0
0

1

4

2
0
Exercise 5. The following table gives the monthly cell-phone usage for the members of three couples.
Name
Aisha
Ava
Binta
Ben
Chen
Camilia
Local Minutes
100
300
50
100
100
200
Long-Distance Minutes
200
0
50
50
300
400
Texts
100
150
0
0
250
200
The major cell phone provider has three different individual rate plans, detailed below.
Name
Lonesome Traveler Plan
Happy Homebody Plan
Chatterbox Plan
Local Minutes
30¢
5¢
5¢
Long-Distance Minutes
30¢
70¢
30¢
Texts
free
2¢
10¢
The major cell phone provider also has three different family rate plans.
Name
Jet-Setter Family Plan
Homestead Family Plan
Gregarious Family Plan
Local Minutes
20¢
5¢
5¢
Long-Distance Minutes
20¢
80¢
20¢
Texts
free
1¢
10¢
Use matrix operations (matrix addition, scalar multiplication, matrix multiplication, transposition) to do the following. Carefully note which operations you’re using. Make sure your
answers make sense!
(a) Make a table of the monthly costs for Aisha, Binta, and Chen under the three individual plans.
(b) Make a table of the monthly demands for the three families (this table will have three rows for the
three couples, and three columns for the family plans).
(c) Make a table of the monthly costs for the three families under the family plans.
(d) Make a table of the monthly costs for the three families under the family plans in dollars (not cents).
(e) (optional!) Make a table of the monthly costs for the three families under all six plans. Choose the
best plan for each family.
2