MA 305 Homework 3

Due 5:00 pm on Monday March 25,  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
 
[  1  0 -3   2 ]
A= [  2  0   2   0 ]
[  0  3 -2   0 ]
[ -3 0   0 -1 ]
  1. Write down the matrix M=x*I4-A where x is a variable.
  2. Compute det(M) using cofactor expansion. First express the determinant as a sum of 3x3 cofactors and then compute the determinant.


2.  Let
 
[ 1 -2 0 ] [ x1 ] [  3 ]
A= [ 0 -1 3 ] x= [ x2 ] b= [ -2 ]
[ 2  4 1 ] [ x3 ] [  4 ]

  1. Without computing A-1 prove that A is invertible.
  2. Using Cramer's rule, express each component of the solution to Ax=b as a quotient of 3x3 determinants.
  3. Compute each of the determinants from part b to find the unique solution to Ax=b.
3. Let V=R2 be two dimensional column vectors with the normal addition and scalar multiplication defined as follows.
 
[ u ] [ (1/a)*u ] [ u ] [ 0 ]
a * [ v ] = [ (1/a)*v ] when a != 0 and 0 * [ v ] = [ 0 ] .

 

Prove or disprove, (V,+,*) is a vector space.

4. Let A be a matrix in Rn x n and let NR(A)={B in Rn x n such that AB=0n x n }. Prove that NR(A) is a subspace of Rn x n .

Bonus: Let A be as in problem 4 and define NL(A)={B in Rn x n such that BA=0n x n }. Prove or disprove, NL(A)=NR(A) for every A in Rn x n.