Problem of the Week
- March 22, 2004.
(a) Let G = {M1, M2, ... , Mr} denote the set of n × n
permutation matrices (those matrices which have only zeroes and ones as entries,
and whose row and column sums are all one),
and let A = (M1 + M2 + ... + Mr) / r denote the average of these matrices.
Determine A.
(b) Let G = {M1, M2, ... , Mr} be any finite set of n × n
matrices with real entries and such that G is a group under matrix multiplication.
Let A = (M1 + M2 + ... + Mr) / r denote the average of these matrices.
What can be said about A?
Standard disclaimer for problem solving: If you feel a problem is posed incorrectly, you should attempt to decide which conditions need to be added, or which modifications need to be added to make the problem an interesting one. Then, solve the revised problem.
Problems from previous weeks
We will have some small reward for the FAU student, graduate or undergraduate, who submits the best solutions to these problems over the next semester. If you have a partial solution, and if it is written well, that may be given some consideration. The problems will be chosen from a variety of sources, which I will not usually identify. I don't intend to post solutions on the website. Solutions should be submitted within one month of the posting date for the problem.
Submit solutions, preferably typed, double-spaced, to me (Dr. Locke).
My colleagues are invited to submit problems for the students. Students may also pose problems. In any case, I will select the problems which appeal to me.
Correct solutions received from students: A. Israel (1), K. Umeda (1), M. Wess (2), Israel-Wess (1-ε).
Problems proposed or communicated by colleagues: SM (1), TPS (1). Thank you.
URL: http://math.fau.edu/locke/courses/ProblemSolving/Problem_of_the_Week.htm