MA-305 Homework 1

Solution

Due at 4:05pm, Tuesday, September 2, 1997

You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

• Find all solutions to the given linear system.
  (a):
  x+ y+2z=-1
  x-2y+ z=-5
  3x+ y+z= 3
  
  (b):
  x+ y+3z+2w= 7
  2x- y +4w= 8
  3y+6z = 8
  
  (c):
  x+2y-4z= 3
  x-2y+3z=-1
  2x+3y- z= 5
  4x+3y-2z= 7
  5x+2y-6z= 7
  

  Solution:

  (a): x=1, y=2, z=-2.
  (b): No solution, the system is inconsistent.
  (c): x=1, y=1, z=0.

• 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
  2x+3y+2z= 5
  2x+3y+(a²-5)z=a+1
  

  Solution:

  (a): If a is a square root of 7.
  (b): If a is not a square root of 7.
  (c): No such a exists.

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

  Solution:

  a-b+c=0

• If
  

  	[ 0	0	-1	2	3 ]
  A=	[ 0	2	3	4	5 ]
  	[ 0	1	3	-1	2 ]
  	[ 0	3	2	4	1 ]

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

  Solution:

  	[ 0	2	3	4	5 ]
  C=	[ 0	0	-1	2	3 ]
  	[ 0	0	0	-7	-14 ]
  	[ 0	0	0	0	4 ]

(Homework resource: Introductory Linear Algebra with Applications by Kolman and Hill)