Math 18.06, Spring 2013
Problem Set #6
April 5, 2013
Problem 1 (5.2, 4). Identify all the nonzero terms in the big formula for the determinants
of the following matrices:
1 0 0 1
1 0 0 2
0 1 1 1
0 3 4 5
,
A=
B=
1 1 0 1
5 4 0 3 .
1 0 0 1
2 0 0 1
For A, we
J =
2
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
,
B ~ w k v fe
yY#B h ggWdS
9 d Bc b ` X V T R PH F D B
W GQ)SP aYW @T USQIGEC' @
@
A
P rxH rQH
F ` wcH d
P Q)xwxYSyxWrQH
F Bc b w Hc w R P o wH y
wD
Yy
P AP T
30
04
@ GIIQW8B x#ExSiGyGI#B xEb
18.06 Problem Set 1
SOLUTIONS
1. Section 2.1, Problem 10
Answer: Ax = (18, 5, 0); Ax = (3, 4, 5, 5).
2. (a) What two vectors are obtained by rotating the plane vectors
1
0
and
0
1
by 30 degrees in the clockwise direction? Write a matrix A such that for
ev
18.06 Problem Set 4
SOLUTIONS
1. Section 3.5, Problem 42
Solution: If the 5 by 5 matrix [A b] is invertible, b is not a combination of the
columns of A. If [A b] is singular, and the 4 columns of A are independent,
b is a combination of those columns.
2.
18.06 (Spring 14) Problem Set 4
This problem set is due Thursday, Mar. 13, 2014 by 4pm in E17-131. The problems are
out of the 4th edition of the textbook. This homework has 8 questions worth 100 points
in total. Please WRITE NEATLY. You may discuss with
18.06 Problem Set 2
SOLUTIONS TO SELECTED PROBLEMS
1. Section 2.5, Problem 25
3 1 1
Answer: A1 = 1 1 3 1 ; the columns of B add up to 0, so B 1
4
1 1 3
does not exist.
2. Section 2.5, Problem 30
Answer: not invertible for c = 7 (equal columns), c = 2 (equ
18.06 (Spring 14) Problem Set 5
This problem set is due Thursday, March 20, 2014 by 4pm in E17-131. The problems are
out of the 4th edition of the textbook. This homework has 8 questions worth 100 points
in total. Please WRITE NEATLY. You may discuss with
Problem Set 3 Solutions:
Section 3.2
6.
1 3 5
0 0
0
The free variables are x2 and x3 so the special solutions to Rx = 0 are (3, 1, 0) and (5, 0, 1).
1 3 0
Elimination on B gives the row reduced echelon matrix R =
0 0 1
The free variable is x2 so the speci
Technology Student Association
2011 National TSA Conference Results
June 21-25, 2011
Dallas, TX
AGRICULTURE AND BIOTECHNOLOGY ISSUES (MS)
Place School Name Team #
State
1
R. Dan Nolan Middle School 1
FL
2
Blacksburg Middle School
1
VA
3
Richland Junior Hi
P iR
) a p
=4
A
P B d e R B H B
rEXAfGGp QH
F P eH t B R P d R P
S)XS)S"S`
@
p
4 4 8
A = 4 4 8
8 8 16
24, 0, 0 det A = 0
Ba
uitr pi g
qsq)hf
1
A
e Bd c a Y W T R PH F D B @
xGQ)SP b`[email protected] USQIGEC' w
@ d RH
a x)A)IQH
nR
X
12
2 13
AT A =
65
56
p B FH R
Set 1: Multiple Choice Questions on Limits and Continuity
3. C
30. A
4. A
31. E
12. C
32. C
14. C
33. B
20. C
34. B
25. D
35. D
28. E
Set 2: Multiple Choice Questions on Differentiation
1. E
2. A
3. B
4. B
11. A
16. A
20. C
21. C
35. C
36. B
41. B
55. B
6
February-March 2014 Issue
Welch Elementary
Chorus honors Dr.
Martin Luther King Jr.,
while senior varsity
football players get
together one last time.
Whats Inside
Page 2 Superintendents Message
Page 3 Elvina Knight to Retire in June
Page 4 Mentoring Gran
*
WARNING!
This is a United States Government computer. This system is for the use of
authorized users only. By accessing and using the computer system you
are
consenting to system monitoring, including the monitoring of keystrokes.
Unauthorized use of, o
Caesar Rodney School District Bi-monthly
Where Excellence is our Tradition
June 2012 Issue
A giant hand-painted mural
stretches from the hallway of
George Welch Elementary
School over to
Dover Air Force Base
Middle School on
Dover Air Force Base.
Whats In
, _
.) , 123 Chapter 4: Applications of Differential Calculus
Set 4: Multiple-Choice
Questions on Applications
of Differential Calculus
WA
AA) -_ W L
gfg; - h" :13? . Dll The slope of me curve y -xy= 4- at the point where y_= 2'13 3 3 f-
\u u. I wh
18.06 Problem Set 7
SOLUTIONS
1. Section 6.1, Problem 2
Answers: Eigenvalues: 1,2 = 1, 5; eigenvectors: v1,2 = (2, 1), (1, 1). The
matrix A + I has the same eigenvectors and eigenvalues increased by 1, i.e.
0 and 6.
2. Section 6.1, Problem 12
Answers: P h
U be)guI sqyV ige7b`D' s
x Xw v d t r h f dc a Y X V
j
j
V dg j `g "`X7add7agin9b)gnc r dw`g
Xv a
XY
X d v fc jX v d a t X u t d f
d XY
fx)ng`YX7aa$dgdso d a t
hw d d w f
d Y ww h X a Y f
dX`g en)fw7X`nc' V dd`Yg p)ng`7ns`Gnd9ec
X f x hw d Y X a fc w
k =1
`
IW
A
l gi ffXR
A x ed
e Ad c a Y W S Q IG E C A 9
@U FP(RI b`XV9 TRPHFDB& X
1
i
Atk
=I+
k!
k =1
tk
k!
A 3 = A4 = = A
1
i
+
1
2
1
i
=
cos t
sin t
A = I + (et 1)A =
et 3(et 1)
0
1
.
Q W
Ak tk
=I+
k!
+ 1 eit
2
1
2
9
k =1
=I+
1
i
=
9
e
At
A2 = A
1
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Course 18.06: Problem Set 1
Due 4PM, Thursday 17th September 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and
Course 18.06: Problem Set 2
Due 4PM, Thursday 24th September 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and
Course 18.06: Problem Set 3
Due 4PM, Thursday 8th October 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and cl
Course 18.06: Problem Set 4
Due 4PM, Thursday 15th October 2015, in the boxes at E17-131.
This homework has 5 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and c