Math 3124 Wednesday, September 3
Second Homework
Due 9:05 a.m., Wednesday September 10
1. Page 34, Problem 5.14. Also, is this group Habelian? (3 points)
2. Recall that the transpose A0of the matrix Ais obtained from Aby interchanging the
rows and columns of A. In particular, the transpose of a b
c dis a c
b d. Let I
denote the identity 2 ×2 matrix (1’s on the main diagonal and 0’s in the two other
entries), and let Gdenote the set of all 2 ×2 real matrices Awith AA0=I. Prove
that Gis a group with respect to matrix multiplication. You may assume that matrix
multiplication is associative, and also the well-known property that (AB)0=B0A0for
all 2 ×2 matrices A,B. (3 points)
3. Page 39, Problem 6.2. (2 points)
4. Can the position
12 14 5 7
10 3 9
8 2 1 11
6 4 15 13
be obtained from
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
by a sequence of moves consisting of sliding one square into the blank space? (1 point)
(4 problems, 9 points altogether)