McMaster University Math 1A03/1ZA3 Fall 2012
Midterm 1
October 4 2012
Duration: 90 minutes
PRACTICE VERSION
Instructors: R. Conlon, A. Contreras, D. Haskell, E. Harper, C.McLean
Name:
Student ID Number:
This test paper is printed on both sides of the page
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Trots
Lineartlgebry
metals
Recycling
Examples
bicycle
q
*
I
340
f
'
the
wheels
:
Let
b
be
U
t
Zt
pedals
number of
the

.
.
.


.
seats
Zbt
:
b
:
pedals
at
Zb
:
handlebars
340
440
b
( l)
220
=
+
:
We only need
u+2t
+
=
,
(2)
+2T
,
(3)
,
=
220
don't need
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
( Sec
3.4 )
A point
A
and
Xo
direction
W
line
Eqnationsoinesvectorefnatit
t
*o
*
N
+
=
ow
t
,
*
parameter

9
direction
position
Parametriceqnatiof
H
*o=
;]
determine
W=[
=/
a
a
we
then
write
5)
*
(5
,
expanding the equation in
/
y
=
xotat
=
+
yo
be
t
coo
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
(
point
OTP
M *
Kot
at
line )
Given
and
line
L
:
*
t N
yo /
Distancing
QIIMM
Disttp
Hmm
HMIP
.
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P
M
=
be
yo
a
a
point
and
a
=
,
Q=*o
direction
orthogonal
P
to
W
/
.4=HPwin#l1/=/0YMm=
The
orthogonal
projection of
along
m
is
Q
Pwjiw (
)
=
mm
H
L
A
(
P
T
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Vectorspacy
( Rough Idea ) A
of objects
collection
called
vectors
YEW
with
two
operations
( )
if
(2)
zero
1
(3)
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( Axiom
1
*
#
Rt
cfw_D
,
4
.
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is
E
number
a
6
.
of
1123
,
,
also
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be
vector
is
C
scalar
and
a
,

then
W
.
,
vector
*
are
space
E
.
(
+
t
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltzsubspay
then
18
4.8
Given
of
set
a
span cfw_ W
,
,
.
.
.
,Wk
=
is
ed
a
cfw_
=
cfw_
C
,
N
subspace of Rn
1123
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x
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.
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axis
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cfw_W
vectors
,
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=
,
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+
"
it
.
.
.
,Wk
in
CKIVK
R
,
in
"
that
where
dene
,
linear
are
C
,
,
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
numbers
Aux
.EE
d Sec 3.1
We
say
Def
.
that
C
,
4
Wv
,
.
.
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P 139
.
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W
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,
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nd
and
K
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0
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in
C
,
.
i.
.
+
.
of
CKNK
ii
of
ck
+
comb
Aux
linear
+
,
comb
bid
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Dene
for
only R3
IUXW
wr
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for
This
a
vectors
dened
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=
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e
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cd
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l
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174
.
a
Theorem
sometimes
,
3.5
.
2
for the
we
call
it
a
e
v
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
#
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.
R
"
numbers
think
we
of
it
vector
a
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a
like
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n
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we
geometric object (
a
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W
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apply
(
a
,
matrix
.
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mu
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Equations of
Same
apply
Line~Y

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line
equation
Ve#m
to
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equation

y
/
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tat
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1
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34
(
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=[N=(
and
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tg
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,
Eqnationofplanett
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
1. Find examples such that
(i) If u v = u w, then v = w.
(ii) If u v = u w, then v = w.
(iii) If A is invertible, then A2 + 2A + I is not invertible.
2. Find examples such that
(i) A is invertible and is diagonalizable.
(ii) A is invertible, but is not di
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
MATH 1Z03/1 3:
Instructors: Hi 11 ' ,
Date: February 25, 2014  Group A @230
Duration: 90 min. erar’t tin/10
Name: ID #:
Instructions:
This test paper contains 23 multiple choice questions printed on both sides of the page.
The questions are on pages 2 th
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
McMaster University Math1ZC3/1B03 Winter 2014
Page 10 of 16
17. For 3x3 matrix A, where
3
0 0
A = 6 6 k
0 3 3
one of the eigenvalues is 0. What is k?
(a) 5
(b) 3
(c) 6
(d) 0
(e) 2
18. For 3x3 matrix A, where
4 2 6
A= 2 1 0
1 0 1
one of the eigenvalues
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltkdynamicalsysay
(
A
probability
theory
in
long
the
)
mm
,lity
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vector
called
is
if
,
all
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i
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and
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=
1
stochasticmatnx/Matkovmatnf
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matrix
if
all
its
A
is
columns
called
are
a
probability
vectors
.
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#
notes
the
to
Suppo
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Now we consider the reverse process of taking power. Given a xed complex number
z = r(cos + i sin ), solve the following equation for w,
wn = z.
I explained in class that there are always n solutions for this equation, which are the nth
roots of z . They
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Oa
@
Compntationofdeterminiants
!1!
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for
Nott
last
covered
product of elementary
)
class
matrices
today
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if
(
expansion
of elementary
B
EA
=
Properties
Adjoint
matrix
and
.
of
Cramers
matrix
compare
det B
with
diet A
Det
Rule
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Remember
(
l
)
(2)
(3)
@
@
(A)
det
A
is
det (A)
A
of
dat
for
=
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if
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the
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,
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of
form
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and A
bc

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general
the
in
=
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; =/
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,
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appears
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
NEW
iplicatoy
Given (
)
mxn
matrix
A
and
(
nxr
)
matrix
B
me
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it
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Invetybematricey
5
9f
A
is
El
matrices
then
,
.
.
.
Ek
,
matrix
nxn
square
a
A
and
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that
such
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,
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there


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it
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.
.
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
statements
The following
( l )
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invertible
is
The
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A
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*
A
'
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matrices
f
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identity
.
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that
such
exists
elementary
product of
a
A
,
of
A
i.e.
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Ex
Gussian/Ganss.JandanEliminaio
a
Sec 1.2
5
li
leading
EIlIEiEt.HE!t*
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He
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ktEtHkdENkecH
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:
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z
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Sec
1.2
leading
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.
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li
non

pivotal
column
tIHHIHIEH
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Invetybematricey
A
9f
is
El
matrices
then
.
.
.
inverse
Ek
,
A
of A
At
=)
*
5
g.
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The
sec
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matrix
A
=
(
,
=
g
and
Ek Ek
if
there


,
.
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exist
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.
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elementary
.
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A
I
=
invertible
is
is
At A
,
that
such
that
say
we
The
,
matrix
nxn
square
a
:=
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
#
Matrix
Operations
when
1st
now
nth
2nd
1st
we
al
fate
ai
,
912

.
ain

li j
) entry
of
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,
m
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rows
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case
.
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an
=
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a
of

.
.
a
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)
vector
now
.
whin
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
\
Two
motivating
Suppose
by
equations
0.52
0.53
0.55
on
ed
two
Suppose
B
controlled
,
X
by
Yi
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of
each
Again
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them
linear
,
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.
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growing
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t
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algebra problem
.
proportion
should
?
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98
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X