Sample Examination 2 Matrix Methods
1.
Find the eigenvalues of the matrix A =
3
2
.
2 6
SOLUTION. The characteristic polynomial of A given by
pA ! =
3! 2
= 3 + ! 6 + ! 4 = ! 2 + 9! + 14 = ! + 2 ! + 7
2 6!
and the eigenvalues, which are the roots of the ch
Discrete Mathematics Final Review
I.
All Previous Review Sheets
I.
Graph
A.
Definition of G = (V, E)
1.
Digraph
a.
Directed edges
2.
Bipartite
3.
Edges
a.
Directed
b.
Paths
i.
Euler path
ii. Simple path
c.
Cycles
i.
Euler Cycle
d.
Loops
e.
Weights
i.
Mult
Discrete Mathematics Review 2
Recursively Defined Objects
A.
Recursive Definition
1.
Basic Step
2.
Recursive Step
B.
Examples
1.
Binary Trees
2.
Rooted Trees
3.
Concatenated Strings
C.
Structural Induction
1.
Induction that works well with recursive defin
Discrete Mathematics Review 1
I.
Logic
A.
Proposition
B.
Logical Operators
1.
And
2.
Or
3.
Xor
4.
Not
5.
Implies
a.
b.
Hypothesis and Conclusion
c.
Truth Table
d.
6.
In terms of Or and Not
Converse, Inverse, Contrapositive
Biconditional
a.
b.
C.
If and on
Discrete Mathematics Review 2
0.
I.
Recurrence Relations (from last review sheet)
A.
Constructing sequences
1.
Towers of Hanoi
2.
Strings
3.
Fibonacci sequence
B.
Difference equations as recurrence relations
a.
Closed form for recurrence relations
C.
Homo
Discrete Mathematics Review 2
I.
Mathematical Induction
A.
Steps in mathematical induction
1.
P(1)
2.
P(n) P(n + 1)
B.
Strong Induction
1.
uses all the P(m) for m < n
C.
Well-Ordering Principle
1.
Minimal element for which a property is not true
II.
Recur
Examination 2 Solutions Discrete Mathematics
1.
Are the following pair of graphs isomorphic? WHY?
u1
u2
u3
u5
u6
u4
u8
u7
v1
v2
v4
v3
v5
v6
v8
v7
SOLUTION. The graphs are not isomorphic. One way of seeing this is to note that each graph has precisely two
Solutions Discrete Mathematics Examination 2
1.
Determine whether the relation with the following graph is an equivalence relation.
a
b
c
d
SOLUTION. The graph does not represent an equivalence relation since (c, a) and (a, d) are edges but (c, d) is not
Solutions Examination 2 Matrix Methods
1.
Find the characteristic polynomial and the eigenvalues of the matrix A =
2 0 -1
121.
-1 0 2
SOLUTION. The characteristic polynomial is
cA HxL = Det
2-x
0
-1
2 - x -1
1
2-x
1
= H2 - xL Det
-1 2 - x
-1
0
2-x
= H2 -
Review Sheet 2 Matrix Methods
I.
II.
Determinants
A. Three Defining Properties
1. Det I = 1
2. Multilinear
3. Anti symmetric (interchanging two rows produces a minus sign)
B. Evaluation by Cofactors
1. Expansion along any row or column (to take advantage