Math 2270 Assignment 10
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Dylan Zwick
Fall 2012
Section 5.2 1, 3, 11, 15, 16
Section 5.3 1, 6, 7, 8, 16
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-
1
5.2 Permutations and Cofactors
-
1 Are their rows
5.2.1 Compute the determinants of A, B, C from six terms.
independent?
3\
/1 2
A=(312)
2
/1
2 3\
B=14441
1)
6
-
‘Using the big formula.
2
1
1
c=fiio
\i
00
‘a
IAH I. I±t3-
I
7)
/1
5.2.3 Show that det(A)
=
0 regardless of the five nonzeros marked by x’s:
A=
(
0
0 r
0 x
What are the cofactors of row 1?
What is the rank of A?
What are the 6 terms in the big formula for det(A)?
o c
OyJ
—
/0
-
0
00
Oc
01
lie #Iirs o
(A)
3
co/mTh
cs
U
E
0
U
U
0
-d
(J:
0
U
8
c%i
L
II
ij
(J
cc
H
I.’
H—
5.2.15 The tridiagonal 1, 1, 1 matrix of order
n has determinant E:
E2=
1100
1110
0 1 1 1
0011
110
==111
3
E
011
1
(a) By cofactors show that E = E_
—
.
2
E_
2 = 0 find E
,E
3
,.
4
1 = 1 and E
(b) Starting from E
.
,
.
8
E
100
E
(c) By noticing how these numbers eventually repeat, find .
014
CCGY
0
(
row
I 1 00UI I-Q
o i H
n
ijq4
QW?1
CQ((
i--f-c)
-e
k1
I
E
1
5)
Ei
a
(0
tcIot
5.2.16 F is the determinant of the 1, 1, —1 tridiagonal matrix of order n:
1
1
F
1
=
2
1
==
4
F
1
1
1—10
F
=
3
1 1-1=3
01
1
—1
1 —1
1
1 —1
1
1
1 + F_
. These determi
2
Expand in cofactors to show that F = F_
The sequence ususally
nants are Fibonacci numbers 1, 2, 3, 5, 8, 13
starts 1, 1, 2, 3 (with two l’s) so our F is the usual F+
.
1
1 -1
o
i
o
-
-
-
0
1-io
o
11
HI
+
I
6
Q(\
cr
E
—
crJ
)
—
—
—
II
JD
-
.
ct
(rJ
)
‘S
L
U-C
Lt)
II
II
++
lw
CD
+ +
++
‘S
—
—
II
DlrcN
r-J
0—
It
rJ
><_
0
—
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)
r’J
j
1
(N
-c
(N
N
Ii
(N
\Jjf%)
‘J
-“J
N
c)
(N
2:’
(J
[I
-
H
—I
\jj
H
H
JO
c)
-
(N
J)
(N
C-,
(N
(N
1
çiJ\J
N
(N
CD
CIT
CD
(ID
CT:,
H
C)
C
C
(Th
n
C
CD
0
‘-1
0
5.3.7 If all the cofactors are zero, how do you know that A has no inverse?
If none of the cofactors are zero, is A sure to be invertible?
0/ h/2
T
1
*r
A
fie
A
/
C
41
e 4)
0
(5
/
I
A
/
1,
2)
/
c,
/
All
A
/
4d -e4C
1 C/
‘I
5.3.8 Find the cofactors of A and multiply ACT to find det(A):
/1 14
A=f1 22
25
3)
and
and ACT
If you change that 4 to 100, why is det(A) unchanged?
C5-3
(
(z- iJ-/
-5Q_
I—
/
(ZOt
32
f3
(4)
-
C
o1
/ H
/ ?
(
6
—
I
e#(A) 3
(J5Q
Ct
1oy
COTQ cot
(()
i(-3)
36)0
C) 30
0 03
3
+
10p)
be
c
,
t/:
5.3.16 (a) Find the area of the parallelogram with edges v
w= (1,4).
=
(3, 2) and
(b) Find the area of the triangle with sides v, w, and v + w. Draw it.
(c) Find the area of the triangle with sides v, w, and w
3
A reck
)2)
1 ‘-1
1
—
v. Draw it.
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(lo)
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11
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