MA 305 Homework 2

Due 11:59 pm on Friday March 8,  2002

Calculations necessary for these problems may be done either by hand or with Maple. Solutions are to be submitted as a ASCII text (.txt) or Maple Worksheet (.mws) (version 6 or lower),  file via WolfWare.  If you must use Maple 7 then submit your work  in maple text or plain text format, Maple 7 crashes the grader's Maple 6.

If you use Microsoft word, please convert your file to regular text with a .txt extension before submitting the file.

Submitted files should have either .txt or .mws extensions. This makes the grader's life much easier.

Show your work for full credit. This means show all of your maple commands or if done by hand show most of the intermediate steps so we know how you arrived at your answer.

You must write up your own homework.  You may work together in the formulation of your answers, but you must write up your answers separately.

Put your name and the name of all the people you worked with at the beginning of your submission file.

Questions about the assignment may be sent to the TA, George Yuhasz at  gyuhasz@math.ncsu.edu

1. Let
 
[ 2 -4   1 ]
A= [ 1   0 -2 ]
[ 0   3   0 ]
Compute A3-A and -A2+A-4*I3.

2.  Let
 
[ 1 1 -3 ] [   1   6 -3   3 ]
A = [ 2 0 -1 ] and  B = [ -4   5 -2   7 ] .
[    3 -1 -1   2 ]

Compute each of the following if possible and if not possible explain why.

  1. (AB)T
  2. ATBT
  3. AAT
  4. BTB
  5. BTAT


3.  Find A-1 using the Gauss-Jordan elimination method discussed in class (do not use the matrix inverse command in Maple) if
 
[   0 -1 -2 ]
A= [   0   4    4 ]
[ -2   2    0 ]

4. Let
 
[ -1 1   2 ]
A = [ -2 2   2 ]  . 
[ -1 1 -2 ]

  1. Compute  3x3 matrices T and U such that TA=U and U is in row echelon form.
  2. Write T as a product of elementary matrices.
  3. Use T and U to solve the systems of equations Ax=b1 and Ax=b2 where
    4. [ x1 ] [ 3 ] [   3 ]
    x = [ x2 ] b1 = [ 5 ] b2 = [ 10 ]  .
    [ x3 ] [ 1 ] [ 12 ]