MATH 631 Homework for Chapter 1 September 3, 1999

Read Chapter 1 and Sections 1 & 2 of the Appendix.

The following problem is to be handed in (for a grade) on Monday, September 13. There may be additional problems assigned in a later class.

Let n and m be positive integers. Let A_n be the $n\times n$ matrix with 1's on and above the diagonal, and zeroes below the diagonal. Thus the i,j entry of A_n is 1 if $j\ge i$ and if j<i.

For example, $A_3=\begin{pmatrix} 1&1&1\\ 0&1&1\\ 0&0&1\end{pmatrix}$.

Compute (A_n)^m explicitly. (For example, you may give a formula or expression for the i,j entry for each i,j with $1\le i,j\le n$.) Prove your answer is correct.

For example, $(A_3)^2=\begin{pmatrix} 1&2&3\\ 0&1&2\\ 0&0&1\end{pmatrix}$.If we set B=A₃², we can express this by saying b_ij=0 if i>j and b_ij=j-i+1 if $j\ge i$.

Note: I would like you to do two things.

1.

Give me some idea of how you got your answer. Guessing is fine, as long as you give me an idea how you arrived at your guess.

2.

Give a rigorous proof that your answer is correct.

For part (1), the ``guessing'', here is what I would really like you to do. Use the computer algebra program MAPLE to compute A_n^m for a few n,m, and try to see the pattern.

To use MAPLE, you will want to do something like the following. At the > prompt, enter
A := matrix([[1,1,1],[0,1,1],[0,0,1]]);
evalm(A^2);
This computes A₃². Some important things to notice: First, every command line ends with a  ;   and second, to set one thing equal to another, use the command := (notice the :).
Experiment with some values of m,n, and print out a copy of the MAPLE worksheet.

One other MAPLE command you may want to use:
binomial(a,b);  gives you the binomial coefficient a choose b.