MA 405 Homework 2 DUE: Wed, Mar 11, in class

This assignment will be turned in on paper.

If you use Maple to solve some of the questions, then simply print your worksheet out and attach to it the problems that you did by hand.

Please clean up your Maple worksheets and clearly label the problems. Remember you can omit output on commands by using a colon e.g. with(LinearAlgebra):

1.

Let , then calculate and is a constant.

2. Let and

Calculate the following. If the multiplication is not possible explain why.

AB

A^TB

B^TA^T

(AB) ^T

A^TB^T

(B^TB) ^-1

BA

What can you say about (AB) ^T , A^TB^T , and B^TA^T

Notice that (B^TB) ^-1 exists, but does (B^TB) ^-1 = B^-1 (B ^T)^-1 ? Is B invertible?

3. Using the Gauss-Jordan method (not using Maple) find the inverse of A, where.

4. Consider the finite difference matrix and ,

a) Calculate using the cofactor expansion method.

b) Using Cramer’s rule, find the solutions to Ax=b for each of the components of x.

Please give both determinantal expressions for t,u,v,w and then compute the values of the given determinants.

BONUS: Let where . Find a formula for A^-1 , using the cofactors of A, and give one equation involving a,b,c,d that must be satisfied so that the inverse exist.