MATH 631 Homework for Chapter 1   September 8, 1999

The following 3 problems are to be handed in on Monday, September 13.

Problem A. This is the problem you were already assigned: compute A_n^m in simple closed form , where A_n is the $n\times n$ upper triangular matrix with all entries above and on the [main] diagonal equal to 1.

Problem B. Let $A=\begin{pmatrix} 1&1\\ 0&1\end{pmatrix}$ and let $B=\begin{pmatrix} a&b\\ c&d\end{pmatrix}$ be a ``random'' matrix. Find conditions on a,b,c,d that guarantee

1.

AB=BA;

2.

$AB\ne BA$.

Based on your answer, how likely is it that AB=BA? Is it typical, atypical, can you associate a probability to it?

If we replace A with another matrix, the ``chance'' that AB=BA may change. For example, if A=0 or A=I (the $2\times 2$ identity matrix), then AB=BA always holds. What other matrices A can you find so that AB=BA is more likely than in the case $A=\begin{pmatrix} 1&1\\ 0&1\end{pmatrix}$?

Problem C. (i) Let A and B be matrices. Show that AB and BA are both defined iff [if and only if] both products are square matrices. (This is easy: don't make it hard.)

(ii) If C is a square matrix, we define the trace of C, denoted $\operatorname{tr}C$, to be the sum of the diagonal entries of C. Thus if C is $n\times n$, we have $\operatorname{tr}C=\sum_{i=1}^n c_{ii}$.

Suppose A,B are matrices such that both AB,BA are defined.
Prove that $\operatorname{tr}AB=\operatorname{tr}BA$.

The following problems are meant to give you more practice with the concepts at the start of the book. They do not need to be turned in now. (I may ask you to hand in some of them later.) You are encouraged to use a computer algebra system (i.e., MAPLE) to assist you whenever possible.

Chapter 1, pp. 31-37

§1 #1,4,10,12,15,19

§2 #2,some of 5,8,12,13,14,16,17

§3 #1,2,3,6,8

§4 #1,2,5,6

§5 #1,2,3

Miscellaneous: #3,7

Appendix, p. 599

§1 #1,2,3,4

§2 #2,3,4