Calculations
necessary for these problems may be done either by hand or with Maple.
Solutions are to be submitted written or printed on paper in class. If you turn in your maple file please
clearly label the problems and make them neat.
1. Solve the following systems of linear
equations for the unknown variables.
a.
2x + y + z = 5
4x – 6y = -2
-2x + 7y + 2z = 9
b. u + v + w = 3
2u + 2v + 5w = 2
4u + 6v + 8w = 1
2. Write the augmented matrix for the following system, and then reduce it to a
row echelon form. Next, find all the values of a and b for which the resulting
linear system has (a) no solutions, (b) a unique solution, and (c) infinitely
many solutions.
3x - y + 2z = 1
-x + 2 y + 4z = 3
2x + y + (a)z = (b)
3. Let
the matrix A be an augmented matrix for a system of linear equations where the
first three columns correspond to the variables x, y, and z and a,b,c are real
numbers .
a)
Find
conditions on a, b, and c such that the matrix is in row echelon form.
b) Next, perform Gaussian
elimination to get the row echelon form, and then find conditions on a, b, and
c that are sufficient for the system to be consistent.
[ 1
0 1 : 0 ]
A= [
a b -b : c ]
[ 0
-a a : 0 ]
4. Write the augmented matrix for the
following linear system, and then reduce the matrix to row echelon form. List
the pivots for the system and tell which variables are dependent and which are
free. Finally, obtain the general solution for the system in terms of the free
variables.
u
+ 3v + 3w + 2y = 1
2u + 6v + 9w + 5y = 5
-u -3v +3w = 5