MA-305 Homework 1

Due at 2:35 pm, Thursday, January 22, 1998

Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text, Maple Worksheet (.mws), html, or postscipt-formatted document to the TA, John Haws (jchaws@eos.ncsu.edu).

1. Find all solutions to the given linear system.
   1.   x + y + 2z + 3w = 13
        x - 2y + z + w = 8
        3x + y + z - w = 1
      
      
   2.   x + y + z = 1
        x + y - 2z = 3
        2x + y + z = 2
      
      
   3.   2x + y + z - 2w = 1
        3x - 2y + z - 6w = -2
        x + y - z - w = -1
        6x + z - 9w = -2
        5x - y + 2z - 8w = 3
      
      

2. Find all the values of a for which the resulting linear system has (a) no solution, (b) a unique solution, and (c) infinitely many solutions.
   x + y + z = 2
   x + 2y + z = 3
   x + y + (a²-5)z = a
   
   
   

3. Find an equation relating a, b, and c so that the linear system
   x + 2y - 3z = a
   2x + 3y + 3z = b
   5x + 9y - 6z = c
   
   
   is consistent for any values of a, b, and c that satisfy that equation.

4. If

   A =

   [ [ [ [

   1 2 1 1

   -2  -3   3   1

   0  -1   2   0

   2  5  5  2

   ] ] ] ]

   
   find a matrix C in reduced row echelon form that is row equivalent to A.