Some Maple Warmup Problems
(Math 221 Version)

1. Use Maple to calculate A³ v₁ + 3 v₂ where

$$A = \left( \begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & ... ... v2 = \left( \begin{array}{rrr} 5\\ 7 \\ 8 \end{array}\right).$$

2. Let A be the matrix

$$\left( \begin{array}{rr} .96 & .03 \\ .04 & .97 \end{array}\right),$$

Calculate A¹⁰⁰,A²⁰⁰, and A³⁰⁰. Compare.

3. Use the row operations package to solve the system of linear equations:

$$\begin{eqnarray*} 3 x + y + 4 z & = & 7 \\ x + 2 y + 3 z & = & 8 \\ x - y - z & = & 9 \end{eqnarray*}$$

( Samples using the row operations package are located in the file :Maple V Release 4:Math 221: Row Operations Examples on each Macintosh. )

4. Now use Use Maple's solve command on the above system. (This is easier, but probably less instructive ....) There is also a linsolve command in Maple which could also do this in one step.

5. For each of the following matrices A, describe all 2 x 2 matrices B so that A B - B A = 0:

a)

$$\left( \begin{array}{rr} 2 & -1 \\ 3 & 5 \end{array}\right)$$

b)

$$\left( \begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$$

c)

$$\left( \begin{array}{rr} 3 & 0 \\ 0 & 4 \end{array}\right)$$

d)

$$\left( \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array}\right)$$

e)

$$\left( \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$$

HINT: If U is a matrix of expressions in variables {a,b,c,d}, then solve(convert(U,set),{a,b,c,d}) will generate the solutions to the matrix system U=0.