MA-305 Homework 1 Solutions

# Solutions to Homework 1
# 1.Find all solutions to the given linear system.
# 
# 
# 
#         a.  x + y + 2z + 3w = 13
#             x - 2y + z + w = 8
#             3x + y + z - w = 1
# 
#         b.  x + y + z = 1
#             x + y - 2z = 3
#             2x + y + z = 2
# 
#         c.  2x + y + z - 2w = 1
#             3x - 2y + z - 6w = -2
#             x + y - z - w = -1
#             6x + z - 9w = -2
#             5x - y + 2z - 8w = 3
# Solution
# (a) The solution set is {x = w - 2, y = -1, z = 8 - 2w, w = w}.
# 
# (b) The solution set here is {x = 1, y = 2/3, z = -2/3}.
# 
# (c) Row reduction reveals this system as inconsistent, and thus this
# system's solution is the empty set, { }.
# 
# 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 + (a^2-5)z = a
# 
# (a)   a = (+/-) sqrt(6) provides no solution.
# 
# (b)  For any a <> (+/-) sqrt(6), the system has a unique solution.
# 
# (c)  There is not a real value for a so that the system will have
# infinitely many solutions.
# 
# 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.
# 
# Row reducing the matrix reveals that the parameters a, b, and c must
# satisfy the equation -3a-b+c = 0 in order to be a consistent system.
# 
# 4.If
#                      [ 1    -2     0    2  ]
#                  A = [ 2    -3    -1    5  ]
#                      [ 1     3     2    5  ]
#                      [ 1     1     0    2  ]
# 
# find a matrix C in reduced row echelon form that is row equivalent to
# A.
> 

                            [1    0    0    0]
                            [                ]
                            [0    1    0    0]
                       C := [                ]
                            [0    0    1    0]
                            [                ]
                            [0    0    0    1]

>