MA 305 Homework 2

Due 11:59 pm on Friday March 7, 2003

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 7 or lower), file via WolfWare.

If you use Microsoft word, please save your file as ascii text with a .txt extension before submitting the file.

Incorrectly submitted files will result in NO CREDIT, so be sure the file you submit is readable in a text editor or Maple. If you are using Windows, check text files using notepad to make sure it looks readable.

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 type your own file(i.e. no copying and pasting either).

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

Questions about the assignment may be sent to the TA, Lauren D'Elia at rldelia@math.ncsu.edu.

1. Let

[3 4 1]

A = [2 6 3]

[0 1 8]

Compute A^2-3A and A^3+10I, assuming I is a 3x3.

2. Let

[1 4]

A = [2 5] and

[3 0]

[0 2 1 5]

B = [4 -3 2 -1]

[1 2 0 -2]

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

a.(A^T)A

b.BB^T

c.(AB)^T

d.(B^T)A

e.(B^T)(A^T)

3. Find A^-1 using the Gauss-Jordan elimination method discussed in class (do not use the matrix inverse command in Maple) if

[1 4 3]

A = [-1 -2 0]

[2 2 3]

4. Let

[0 2 3 0]

A = [0 4 5 0]

[0 1 0 3]

[2 0 1 3]

[x1] [8]

x = [x2] b = [3]

[x3] [6]

[x4] [2]

a.Write down the matrix M=t*I-A where t is a variable and I is in 4x4 form.

b.Compute det(M) using cofactor expansion. First express the determinant as a sum of 3x3 cofactors and then compute the determinant.

c. By using Cramer's Rule, express each component of the solution to Ax=b as a quotient of 4x4 determinants.

d. Compute each of the determinants from part b to find the unique solution to Ax=b.