Math 265
Final Exam
Name_______________________________
I. Write T for true or F for false in the box to the right of each question. (20 pts)
1. In a group, closed under multiplication, every element in the group has a multiplicative inverse.
2. In 2-D, the determinant of a reflection matrix about y = 0 has a value of 1.
3. In 3-D, any rotation matrix about the x-axis has a determinant of 1.
4. If A is an invertible matrix then DET(A) = -DET(A-1).
T or F
5. All square matrices are invertible.
6. Every group has an identity element.
7. The following illustrates the commutative property of addition for three elements a,b, and c of set
S: (a+b)+c = a + (b+c)
8. The equation y = x defines a 2-D subspace of R3.
9. Let A be a matrix, v be a vector, and 0 be the zero vector, so that Av = 0. Then v = 0.
10. If Av = 0 with matrix A, and vectors v and 0, then the dimension of v = the dimension of 0.
II. Write BRIEF answers to the following questions.
1. Which matrices, if any, must be added to close the
 1 0   1 0  
set S = , 
,
  under matrix
 0 1  0 1 
multiplication.
You DO NOT have to show your work. (24 pts)
2. Write the 2 x 2 matrix that is the product of a
reflection through y=0 on the left and a 60o
counterclockwise rotation on the right.
3. Write a 3 x 3 matrix that is precisely a reflection
through y = x (left) and y=0 (right).
4. Write a 3x4 elementary matrix A that adds row 2 to
row 3 in some compatible matrix B.
1
0
5. Find the null space of 
0

0
6. State the product of the two matrices:
x x
 y y

1 0 1 0 1  

.
z
z
0 1 0 1 0 



 w w
 z z 
1 1 1
1 0 1 
0 1 0

1 1 1
III. Prove or disprove the following. (30 pts)
1. The set of all 2x2 rotation matrices is closed under
multiplication.
2. A 3x3 rotation matrix about x-axis multiplied by
its inverse gives the identity.
3. The set {1,ex, e-x} is a linearly independent set,
when each function is defined on the set of real
numbers.
 0   1   1 


4. The set  0  ,  1  ,  0   is a basis for R3.
     
 1   1  1 
      
5. The set S = {(x,y) : x ε R and y = 5} is a subspace
of R2.
6. The set {sin2x,10cos2x, 4}is a linearly independent
set of three functions defined on the set of real
numbers.
IV. Solve the following problems.
1. Find the coordinates of a vector with respect to basis V if coordinates of some vector with respect to basis
vector U are (1,1,1). SHOW YOUR WORK, FROM THE DEFINITION OF BASIS THROUGH
COMPLETION. (6 pts)
 1  0  1  
 1   0  1  
      


U =  1 , 0  , 0   V =   0  , 0  , 1  
 1 1   0  
 1  1   0  
      
      
.
2. Calculate the inverse of the matrix below by hand and show your work! Be sure to use the proper “code”
between steps. If the matrix does NOT have an inverse, state why not. (2 pts)
1 0 0 0 
1 1 0 0 

.
1 1 1 0 


1 1 1 1 
3. Consider the traffic flow model below. (18 pts)
1. Find the solution, if it exists, AND INTERPRET your solution. If the solution does not exist, state
both mathematical and physical reasons why the solution cannot exist.
2. Find a null space solution and Interpret your solution.
10cars/hr
x
a) Write the system of equations here.
30cars/hr
20cars/hr
w
y
z
20cars/hr
b) Write the vector solution here.
c) Briefly but clearly explain how the vector solution
makes sense within the physical problem.
d) Write the null space solution here.
e) Briefly but clearly explain how the null space
solution makes sense within the physical problem.
f) Suppose the variable u replaces the “20 cars/hr” input between variables y and z in the above model. Solve
the system in terms of u, write the vector solution below, and clearly interpret the solution in terms of the
variable u.
non-homogeneous solution
interpretation of the non-homogeneous solution in terms of u
g) As in part e, find the null space (homogeneous) solution of the above model in terms of u and clearly
interpret it.
homogeneous solution
interpretation of the homogeneous solution in terms of u