Homework 1
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Friday, February 8.
1. Consider the following matrix, whose entries are elements of the field Z11 .
7 4
A =
4 3
(a) Find the eigenvalues of A. Your answers should be elements of Z11 .
(b) Find one eigenvector for each eigenvalue. Again, the entries of your eigenvectors
should be elements of Z11 .
2. Let p be a prime number.
p
(a) Prove that the binomial coefficient
is a multiple of p for 0 < k < p.
k
(b) Use the Binomial Theorem to prove that
(x + y)p ≡ xp + y p
(mod p).
for all x, y ∈ Z.
(c) Use induction to prove that
(x1 + · · · + xn )p ≡ xp1 + · · · + xpn
(mod p).
for any n ∈ N and any x1 , . . . , xn ∈ Z.
(d) Use part (c) to prove that
np ≡ n (mod p)
for any n ∈ N. This result is known as Fermat’s Little Theorem.
3. Consider the following graph, with vertex set V = {1, 2, 3, 4, 5, 6}.
1
3
2
4
5
6
(a) This graph has four automorphisms. List them, making sure to indicate the image
of each vertex under each automorphism.
(b) Make a table showing the compositions of the four automorphisms you found in
part (a).