PROBLEM SET
Assignment 1
Math 5316, Fall 2012
Sept 20, 2012
• Write all of your answers on separate sheets of paper.
You can keep the question sheet. You can attach
printouts of work done in Maple.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This problem set has 8 problems.
Good luck!
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coefficent matrix in reduced row
echelon form. Determine if the system is consistent and, if it is consistent, find
all solutions.
A.






B.






C.
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
2
−5
0
0

1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
2
−5
0
17













1 0 −2
3 0
0
 0 1 −1 −1 0 −2 


 0 0
0
0 1
3 
0 0
0
0 0
0
Problem 2. In each part you are given the augmented matrix of a system of
linear equations, with the coefficient matrix in row echelon form
(but not reduced row echelon form). Use backsubstitution to find all of the
solutions.
A.
B.


1
2

 0

0
1
0
−1
3


2 5 

1 1
1
1
2
−1

 0


 0

0
0
1
2
0
0
0
0
0
0
5
1


3 2 


1 1 

0 0
Problem 3. In each part, you are given a system of linear equations. Form the
augmented matrix of the system and perform row operations to
1
put the coefficient matrix in RREF. You can use Maple, but show the indvidual
row operations (check yourself with Maple). Hint: When the left hand side is
the same, you can do several systems at once by forming an augmented matrix
with all of the right-hand sides to the right of the bar.
A.
3 x1 + 22 x2 + 5 x3 = −55
3 x1 + 29 x2 + 6 x3 = −75
−4 x1 − 34 x2 − 6 x3 = 88
2 x1 + 20 x2 + 4 x3 = −52
B.
3 x1 + 22 x2 + 5 x3 = 174
3 x1 + 29 x2 + 6 x3 = 221
−4 x1 − 34 x2 − 6 x3 = −256
2 x1 + 20 x2 + 4 x3 = 152
C.
9 x1 + 21 x2 − 39 x3 + 6 x4 + 2 x5 = −36
2 x1 + 8 x2 − 12 x3 − 2 x4 + x5 = −13
2 x1 + 9 x2 − 13 x3 − 3 x4 + x5 = −15
5 x1 − 13 x2 + 3 x3 + 28 x4 − 2 x5 = 20
Problem 4. Consider the linear system whose augmented matrix is


1 2 1 2
 1 1 b x 
1 3 2 1
Determine the values of b for which the system has a unique solution. Find
this solution (in terms of x).
For the values of b such that the system does not have an unique solution,
what values of x make the system consistent? Find all solutions of the system
in this case.
Problem 5. Consider the linear system

2

 −1


A=
 2

 5

2
2
Ax = b where

3 1

1 0 


1 1 


0 2 

−3 1
To determine the vectors b for which the system is consistent form the augmented matrix


2
3 1 b1
 −1
1 0 b2 


 2
1 1 b3 


 5
0 2 b4 
2 −3 1 b5
where the bi ’s are variables. Perfom row operations (you can use Maple) to
reduce the coefficent matrix to Row Eschelon Form. Find equations that must
be satisfied by the bi ’s for the system to be consistent.
Give examples of nonzero vectors b so that the system is consistent, and is
not consistent.
This problem is easier in Maple, especially if you use the partial reduction
procedure that can be downloaded from my website.
Problem 6. Let H be the set of 3 × 3

1
H = 0
0
matrices of the form

x y
1 z ,
0 1
where x, y and z are real numbers. Note that the 3 × 3 identity matrix is in H.
A. Find the formula for the product of two matrices in H. Show that H is
closed under matrix multiplication.
B. Using the previous part of the problem, solve a set of equations to find the
inverse of a matrix in H. Show that H is closed under matrix inverse.
C. Show that H is a group under matrix multiplication. This is the three
dimensional Heisenberg Group. Is this group abelian?
Problem 7. A matrix is said to be upper triangular if the entries below the
main diagonal are zero. For example,


1 2 3 4
0 1 2 3


0 0 1 2
0 0 0 1
is upper triangular.
A matrix A is upper triangular if and only if aij = 0 for i ? j.
Show that the product of two upper triangular matrices is upper triangular.
3
Problem 8. Find two square matrices A and B so that A 6= 0, B 6= 0, A 6= B
and
AB = 0.
Can A or B be invertible?
4