Appendix 1: The Rules of Arithmetic.
Let a, b, and c be any real numbers.
(1) Commutative property of addition a + b = b + a
(2) Associative property of addition
( a + b ) + c = a + ( b + c )
(3) Commutative property of multiplication ab = ba
(4) Associative property of multiplication
( ab ) c = a ( bc )
(5) Distributive property a ( b + c ) = ab + ac
(6) Zero product property
If ab = 0 then a = 0 or b = 0 .
(7) Additive identity a + 0 = a
(8) Additive inverse
Given a there exists a number b such that a + b = 0 .
1

(9) Multiplicative identity a · 1 = a
(10) Multiplicative inverse
Given a = 0 there exists a number b such that ab = 1 .
2

Appendix 2: Formulas for Matrix Addition and Multiplication.
a b
Let A = and B = e f
.
c d g h
The sum A + B is given by the formula:
A + B = a b c d
+ e f g h
= a + e b + f c + g d + h
The product A · B (or just AB ) is given by the formula:
AB = a b c d
· e f g h
= ae + bg af + bh ce + dg cf + dh
3

Worksheet 1: Distributive Law.
(1) Introduce yourself to a couple of your neighbors. Practice the following products by doing them in your head. Discuss with each other how you computed them.
(a) 30 · 89
(b) 60 · 59
(c) 80 · 88
(d) 40 · 109
(e) 24 · 52
(f) 72 · 51
(2) Compute the following sums:
(a) 4
1
13
+
2
17
+ 9
1
13
+
1
17
(b) 4 4
111
− 8
197
+ 4 10
197
− 5
111
(3) The distributive property says that 4(7 + 2) = 4 · 7 + 4 · 2. How does the following array illustrate that fact?
7 7 7 7
2 2 2 2
(4) Do you know of other number tricks that illustrate the distributive law? If so, discuss how these number tricks work.
(5) Consider the “usual” algorithm for multiplying two numbers, like 713 × 6. Can you use the distributive law to explain why this algorithm works?
(6) There’s a trick for computing squares of numbers that end in 9. Here’s the idea
(19)
2
= 20
2
− 2 · 20 + 1 = 400 − 40 + 1 = 361 and
(79)
2
= 80
2
− 2 · 80 + 1 = 6400 − 160 + 1 = 6261 and so on. What’s the pattern? What does this have to do with the Distributive
Property?
4

Worksheet 2: The Zero Product Property.
Form a group with at least one other person. Take turns answering the following questions verbally . Be as precise as possible.
(1) Explain why the following product is nonzero:
7
11
+
√
2 2
100
+ 3
99
(2) Explain why the following product is nonzero:
1
123
1
+
234
1
345
1
+
456
(3) Use the following equation
· 24 −
√
576 = 0 to compute the value of 24
2
.
1
19
(4) Use the following equation
1
53
17
+
999 to compute the value of 77
2
.
· 77 −
√
5929 = 0
(5) Use the following equation
11 ·
1
271
−
823
223033 to show that (271) · (823) = 223033.
= 0
(6) Use the following equation
√
3 + 77 ·
1
17311
−
19864
343865704 to show that (17311) · (19864) = 343865704.
= 0
(7) Which of the following two questions is easier to solve? Explain why.
(a) If ( x − 4)( x − 2) = 0 then what are the possible values of x ?
(b) If ( x − 4)( x − 2) = 1 then what are the possible values of x ?
5

Worksheet 3: Other rules of arithmetic.
(1) Add these numbers as quickly as possible:
12 + 7 + ( − 4) + 0 + ( − 12) + 4 + 25 + ( − 7)
Which rules of arithmetic did you use in your computation?
(2) Compute the following product as quickly as possible:
7 ·
1
11
· 99 ·
13
· 1 ·
9
9
· 11 ·
1
13 7
Which rules of arithmetic did you use in your computation?
(3) Compute the following product:
11
23
19
·
101
·
23
77
·
77
19
·
101
11
Which rules of arithmetic did you use in your computation?
(4) Compute the following sums:
(a)
1
1 −
2
+
1
2
−
1
3
+
1
3
−
1
4
(b)
1 −
1
2
+
1
2
−
1
3
+
1
3
−
1
4
+
1
4
−
1
5
+ · · · +
1
107
−
1
108
+
1
108
1
−
109
(5) As a group, design a couple of arithmetic problems which target some of the rules of arithmetic.
6

Worksheet 4: Matrix Addition and Multiplication.
Compute the sum A + B in each part.
(1) A =
3 7
0 8
B =
1 1
1 2
.
(2) A =
1
0
3
− 2 0
B =
0
7
11
0 − 5
.
(3) A =
2 3
5 7
B =
0 0
0 0
.
(4) A =
1 − 2
3 − 4
B =
− 1 2
− 3 4
.
Compute the product AB in each part
(5) A =
3 0
0 8
B =
1
0
0
− 1
.
(6) A =
1 2
3 4
B =
7 7
7 7
.
(7) A =
1 1
1 1
B =
1 − 1
1 − 1
.
(8) A =
11 17
79 243
, B =
1 0
0 1
7

(9) A =
11 17
79 243
, B =
0 0
0 0
(10) Does there exist a matrix C such that A + C = A for all matrices A ?
(11) Does there exist a matrix D such that A + D = D for all matrices A ?
(12) Does there exist a matrix E such that A · E = E for all matrices A ?
8

Worksheet 5: Identities and Inverses.
Definition 0.1.
Let A be a 2 × 2 matrix. We say that B is the additive inverse of A if:
A + B =
0 0
0 0
(1) Find values for b and c such that the following equation is true
2
− 3
11
7
+
− 2 b c − 7
=
0 0
0 0
Is the solution unique?
(2) Find values for b and c such that the following equation is true
1 1
1 1
+
− 1 b c − 1
=
0 0
0 0
Is the solution unique?
(3) Find values for a, b, c, and d such that the following equation is true
√
37 127
+ a b
=
0 0
19 − 11 c d 0 0
Is the solution unique?
(4) Does every matrix A have an additive inverse? If yes, how would you prove this? If no, find a counterexample.
Definition 0.2.
Let A be a 2 × 2 matrix. We say that B is the multiplicative inverse of A if:
A · B =
1 0
0 1
9

(5) If possible, find values for b and d such that the following equation is true
2 0
0 7
·
1 b
2
0 d
=
1 0
0 1
Is the solution unique?
(6) If possible, find values for a and d such that the following equation is true
− 5 0
− 1 2
· a
− 1
5
0 d
=
1 0
0 1
Is the solution unique?
(7) If possible, find values for b and c such that the following equation is true
1 1
1 1
·
1 c b
1
=
1 0
0 1
Is the solution unique?
10

Worksheet 5 1
2
: More on multiplicative inverses.
(8) Find values for a, b, c, and d such that the following equation is true
2 − 3
0 0
· a b c d
=
0 0
0 0
Is the solution unique?
(9) Which of the following matrices can you find a multiplicative inverse? Which of the following matrices can you show does not have a multiplicative inverse?
− 1 0
(a) A =
0 − 1
(b) B =
11 0
0 101
(c) C =
13 0
0 0
(d) D =
1 0
2 3
(e) E =
4 6
6 9
(f) F = a b a b
11

(10) Solve the equation for a, b, c and d .
a b c d
·
1 1
1 1
= a b c d
12

Worksheet 6: The Rules of Matrix Arithmetic.
The goal of this worksheet is determine whether the following “rules” of matrix arithmetic are true or false. Gather evidence by choosing some matrices A, B and C and computing an example. Be prepared to discuss your evidence.
(1) Commutative property of addition
A + B
?
= B + A
(2) Associative property of addition
( A + B ) + C
?
= A + ( B + C )
(3) Commutative property of multiplication
AB
?
= BA
(4) Associative property of multiplication
( AB ) C
?
= A ( BC )
(5) Distributive property
A ( B + C )
?
= AB + AC
13

(6) Zero product property
If AB =
0 0
0 0 then A =
0 0
0 0 or B =
0 0
0 0
.
(7) Additive inverse
Given some 2 × 2 matrix A there exists a 2 × 2 matrix B such that A + B =
0 0
0 0
.
(8) Multiplicative inverse
Given some 2 × 2 matrix A =
0 0
0 0 there exists a 2 × 2 matrix B such that A · B =
1 0
0 1
.
14

Worksheet 7: Zero divisors and Multiplicative Units.
The zero product property fails for 2 × 2-matrices. This introduces a concept which didn’t exist for numbers.
Definition 0.3.
We say that A is a zero divisor if there exists C =
0 0
0 0 such that:
AC =
0 0
0 0
In this worksheet, we’ll explore matrices which are zero divisors.
(1) Determine whether each of the following matrices is or is not a zero divisor. Explain your answer.
(a) A =
1 0
0 1
.
(b) B =
− 1 0
0 0
.
(c)
(d)
(e)
(f)
C
D
E
F
=
=
=
=
3 0
0 7
0 3
7 0
0 3
0 7
3 3
7 7
.
.
.
.
(g)
(h)
G
H
=
=
1 1
0 1
1 1
0 0
.
.
(2) Under what conditions is a matrix of the form a 0
0 d a zero divisor? Under what conditions does such a matrix have a multplicative inverse?
(3) Under what conditions is a matrix of the form a 0 c 0 a zero divisor?
(4) Let A = a
1
0 and B = b
1
0 b
0
2
. Under what conditions on
0 a
2 will the product A · B equal the zero matrix?
a
1
, a
2
, b
1 and b
2
15

(5) If the matrix M is a zero divisor, is it possible that M has a multiplicative inverse?
(If you aren’t sure how to approach this problem, try finding a multiplicative inverse for one of the matrices from the above list.)
(6) If the matrix N has a multiplicative inverse, is it possible that M is zero divisor?
16

Worksheet 8: Video game basics.
For each pair of matrices P and M , do the following three steps:
• Draw a picture of the triangle in the plane which corresponds to P .
• Compute M P .
• Draw a picture of the triangle in the place which corresponds to M P .
(1) P =

0 0 1


0 2 0

, M =
1 1 1


− 1 0 0
0
0
1 0
0 1


(2) P =
 
3 3 4

− 1 1 − 1

, M =
1 1 1
 − 1 0 0


0 1 0

0 0 1
(3) P =

1 1 2


0 2 0

, M =
1 1 1

1 0 0


0 1 5

0 0 1
(4) P =

1 1 2


0 2 0

, M =
1 1 1

1 0 5


0 1 0

0 0 1
17

Worksheet 9: Video game motions.

0 0 1

(1) Let P =

0 2 0

. Draw pictures of:
1 1 1
(a) P .
(b) T
1 , 1
P .
(c) T
3 , 0
P .
(d) T
0 , − 4
P .
(e) T
1 , 2
( T
2 , − 2
P ).
(f) F P .
(g) F ( F P ).
 
(2) Let P =
2 2 3

− 2 0 − 2

1 1 1
(a) P
(b) T
2 , 2
P
(c) F ( T
2 , 2
) P .
(d) F P
(e) T
2 , 2
( F P ).
Draw a picture of each of the following:
18